Vedanta Excel in Mathematics Book 6 Final (2078)

Algebra

EXERCISE 9.6

General Section – Classwork

Let's study the following tricks of multiplication of two binomials.

x×x 2+3 2×3

(x + 2) (x + 3) x2 ... (x + 2) (x + 3) x2 + 5x ... (x + 2) (x + 3) x2 + 5x+ 6

y×y 4–1 4 × (– 1) y2 + 3y – 4

(y + 4) (y – 1) y2 ... (y + 4) (y – 1) y2 + 3y ... (y + 4) (y – 1)

a×a –3–5 (– 3) × (– 5)

(a – 3) (a – 5) a2 ... (a – 3) (a – 5) a2 – 8a ... (a – 3) (a – 5) a2 – 8a + 15

1. Let’s investigate the facts of tricky calculation. Tell and write the products as
quickly as possible.

a) (x + 1) (x + 1) = ............................. b) (x + 1) (x + 2) = .............................

c) (x + 2) (x + 3) = ............................. d) (x + 2) (x – 1) = .............................

e) (a + 3) (a – 2) = ............................. f) (a – 2) (a + 5) = .............................

g) (y – 3) (y – 4) = ............................. h) (y – 4) (y – 2) = .............................

Creative Section - A

2. Simplify. b) a (a + 2) – 1 (a + 2)
a) x(x + 1) + 2 (x + 1) d) a (a + b) + b (a + b)
c) x (x + 2) – 3 (x + 2) f) 2p (2p – 3q) + 3q (2p – 3q)
e) x (x + y) – y(x + y) h) 4a (3a – 5b) –3b (3a – 5b)
g) 3x (x + 2y) – y (x + 2y)

i) p (7p – 2q) + 4q (7p – 2q)

3. Multiply. b) (x + 3) (x + 4) c) (a + 2) (a – 1)
a) (x + 1) (x + 2) e) (b – 5) (b + 5) f) (p + 4) (p – 4)
d) (a – 3) (a – 2) h) (a + b) (a – b) i) (a – b) (a – b)
g) (a + b) (a + b) k) (x – y) (2x + 3y) l) (x – y) (2x – 3y)
j) (x + y) (2x + 3y) n) (5a – 2b) (5a + 2b) o) (x – 3y) (2x + 5y)
m) (3x + 4y) (3x – 4y)

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Algebra

Creative Section - B

4. Multiply. b) (x – 2) (x2 – 2x + 3) c) (x – 3) (2x2 + 3x – 4)
a) (x + 1) (x2 + x + 1) e) (3a + 2b) (2a – 4b – 5) f) (4a – 5b) (3a + 2b – 7)
d) (2a + b) (a + b + 2)

5. a) If x = (a + 2) and y = (a – 2), show that xy = a2 – 4.

b) If x = (p – 3) and y = (p + 3), show that xy + 9 = p2.

c) If a = (2x – 3) and b = (2x + 3), show that ab + 9 = x2.
4

d) If a = (p + q), b = (p – q) and c = q2 – p2 , show that ab + c = 0.

e) If x = (2a – 3b), y = (2a + 3b) and z = 9b2 – 4a2, show that xy + z = 0.

It's your time - Project work!

6. a) Let's write two expressions of the forms ax and bx + c, where a, b, and c
are integers. Find the product of your expressions.

b) Let's write any two binomial expressions of the forms (ax + b) and (cx + d),
where a, b, c, and d are integers. Find the product of your expressions.

9.11 Division of algebraic terms

Let’s study the following illustrations and investigate the rule of division of
algebraic terms.

Example 1: Divide a) 10x2 by 2x b) 15a3b4 by 5ab2

Solution: 10x2
2x
a) 10x2 ÷ 2x = 10x2 ÷ 2x = 10x2 In division of the same
2x base we should subtract
150 × x × x lower exponent from
= 2×x = 10 x2 – 1 higher exponent of the
2 same base.
= 5x
= 5x

b) 15a3b4 ÷ 5ab2 = 15a3b4
5ab2

= 135 × a ×a × a × b ×b × b × b 15a3b4 ÷ 5ab2= 15a3b4
5× a×b×b 5ab2

=3×a×a×b×b = 15a3–1 b4 – 2
5
= 3a2b2
= 3a2b2

Thus, the rule of dividing an algebraic term by another term is :

(i) Divide the coefficient of dividend by the coefficient of divisor.

(ii) Subtract the exponent of the base of divisor from the exponent of the same
base of dividend.

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Algebra

9.12 Division of polynomials by monomials

In this case, each term of a polynomial is separately divided by the monomial.
Let's study the following examples.

Example 2: Divide (a) 10x2 – 15x by 5x (b) 14a3b2 – 8a2b3 by – 2ab
Solution:

a) (10x2 – 15x) ÷ 5x

= 120x2 – 135x
5x 5x
11
= 2x2 – 1 – 3

= 2x –

9.13 Division of polynomials by binomials

While dividing a polynomial by a binomial, at first we should arrange the terms
of divisor and dividend in descending (or ascending) order of exponents of
common bases. Then, we should start the division dividing the term of dividend
with the highest exponent by the term of divisor with the highest exponent.

Let’s learn the process from the following example.

Example 3: Divide (x2 + 5x + 6) by (x + 3)

Solution: Divide x2 by x x2 ÷ x = x (In quotient)

Multiply the divisor (x + 3) by the quotient x

x(x + 3) = x2 + 3x

Subtract the product from the dividend.

x2 + 5x + 6

± x2 ± 3x

2x + 6

Again, divide the first term of the remainder by the
first term of the divisor. Continue the process till the
remainder is not divisible by divisor.

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Algebra

EXERCISE 9.7
General Section – Classwork

Let’s subtract the exponent of base of divisor from the exponent of the same
base of dividend. Tell and write the quotient as quickly as possible.

1. a) x2 ÷ x = ........ b) x3 ÷ x = ........ c) x4 ÷ x2 = ........

d) x4 ÷ x3 = ........ e) a3 ÷ a2 = ........ f) a2 ÷ a2 = ........

g) y4 ÷ y = ........ h) y4 ÷ y4 = ........ i) y5 ÷ y3 = ........

2. a) y2z2 ÷ yz = ........ b) y3z2 ÷ yz = ........ c) y2z3 ÷ yz = ........

d) a3b3 ÷ a2b = ........ e) a3b3 ÷ ab2 = ........ f) a3b3 ÷ a2b2 = ........

g) 6x3 ÷ 2x = ........ h) 6a4 ÷ 3a = ........ i) 12p6 ÷ 4p3 = ........

Creative Section - A

3. Expand the terms of dividend and divisor, then divide.

Eg. 14x4 ÷ 7x2 = 14x4 = 124 × x ×x × x × x = 2x2
7x2 7×x×x

a) x2 ÷ x b) y3 ÷y c) a4 ÷ a2 d) 6x3 ÷ 2x2

e) 6y4 ÷ 3y3 f) 16x5 ÷ 4x2 g) 24x3y4 ÷ 8x2y2 h) 30a4b4 ÷ 5a2b3

4. Divide by using rule.

Eg. 24x5 ÷ 6x3 = 24x5 = 4x5 – 3 = 4x2
6x3
a) x3 ÷ x2 b) a4 ÷ a c) 9p2 ÷ 3p
f) 14x6 ÷ (– 7x5)
d) 15m5 ÷ 5m2 e) 9y6 ÷ (– 3y4)

g) a2b2 ÷ ab h) x3y3 ÷ xy i) 3x5y4 ÷ x2y

j) 10x3y2 ÷ (– 2xy) k) – 12a4b4 ÷ 3a2b2 l) 16x8y7 ÷ (–4x6y5)
5. Divide.

a) (10x2 – 15x) ÷ 5 b) (8x2 – 16) ÷ 8
c) (a2 + a) ÷ a d) (x2 – 3x) ÷ x
e) (4b2 + 6b) ÷ 2b f) (9c3 – 6c2) ÷ 3c2

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Algebra

g) (8m3 – 6m2) ÷ 2m2 h) (6n4 – 9n3) ÷ 3n2
i) (p3q – 2p2q3) ÷ pq j) (3a4b – ab4) ÷ ab

k) (3x3y2 + 6x2y3) ÷ 3xy l) (15p4q3 – 20p2q4) ÷ 5pq

m) (12a6x5 + 18a5x6) ÷ 6a3x3 n) (20b4c6 – 50b4c4) ÷ (–10b4c4)

o) (25x5y4z3 + 40x4y5z4) ÷ 5x3y4z2 p) (21m3n4p5 – 49m6n5p4) ÷ 7m2n3p4

Creative Section - B

6. Find the quotient. b) (a2 + 5a + 6) ÷ (a + 3)
a) (x2 + 3x + 2) ÷ (x + 2) d) (p2 + 9p + 20) ÷ (p + 5)
c) (m2 + 7m + 12) ÷ (m + 4) f) (b2 + 2b – 15) ÷ (b – 3)
e) (a2 + a – 6) ÷ (a – 2) h) (y2 – 4y – 21) ÷ (y – 7)
g) (x2 – 3x – 10) ÷ (x – 5)

7. a) The product of two algebraic terms is 6a3b2. If one of the terms is 3ab,
find the other term.

b) The product of two algebraic expressions is 14x3y2 – 35x2y3. If one of the
expressions is 7x2y2, find the other expression.

c) The area of a rectangle is x2 + 4x + 3 sq. units and its length is (x + 1)
units. Find its breadth.

d) The area of a rectangle is x2 + 3x – 10 square units. If its breadth is (x – 2)
units, find its length.

8. a) If a = 3x2y, b = 4xy2 and c = 2xy, find the value of ab .
c
pq
b) If p = 5a2b2, q = 4a3b3 and r = 10a4b4, show that r = 2ab.

c) If x = 12p3q4, y = 6p2q3 and z = 2p2q2, show that x + y = 2pq + 3q.
y z

It's your time - Project work!
9. Let's write the appropriate terms in the blanks to match the given quotients.

a) ......... ÷ ......... = x b) ......... ÷ ......... = x2 c) ......... ÷ ......... = x3

d) ......... ÷ ......... = 2a e) ......... ÷ ......... = 2y2 f) ......... ÷ ......... = 3p3

10. Let's write any appropriate algebraic terms in the blanks and find the
quotients.
a) ....... ÷ ....... = ....... b) ....... ÷ ....... = ....... c) ....... ÷ ....... = .......

d) ....... ÷ ....... = ....... e) ....... ÷ ....... = ....... f) ....... ÷ ....... = .......

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Unit Geometry: Point and Line

12

12.1 Point, line, ray, line segment and plane - review

Geometry is a branch of mathematics dealing with shape, size and other
properties of different figures. The word ‘geometry’ is derived from the Greek
word ‘geometron’ where ‘geo’ means ‘Earth’ and ‘metron’ means measurement.

In geometry, we study about point, line ray, line segment, plane, angles different
plane shapes, solid shapes, geometrical constructions, etc.

Point AB

It is a mark of a position. R C
It has no length, breadth or height.

It is denoted by capital letters.

In the given figure, P, Q, R, A, B, C, etc. are different points.

Real life examples: A dot made by the tip of a sharp pencil, location of places on

a map, stars in the sky, thumbtack, etc.

Line
It is a straight path which can be extended
indefinitely in both the directions.
It is shown by two arrow heads in opposite
directions.
It can be straight or curved but it is a straight line
when we simply say 'a line'.
It can never be measured because it has no
endpoints.

In the figure, AB and PQ are straight lines. XY is a curved line.
Real life examples: Number line, rubber band while stretching both directions, etc.

Note:
(i) An infinite number of lines can be drawn through a given points.
(ii) One and only one straight line can be drawn through two points.
(iii) An infinite number of curved can be drawn through two points.

Ray

It is a straight path which can be extended indefinitely

only in one direction and the other end is fixed. R
Q
It's fixed end is also called the initial point.
P
The end which can be extended is shown by an

arrowhead. O

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Geometry – Point and Line

In the figure, AB, PQ, OQ and QR are rays.
Real life examples: sun rays, ray of light of torchlight, projector, etc.

Line segment
A line segment is a part of a straight line, It has a
definite length. In the figure, AB and PQ are line
segments. Real life examples: a piece of wire,
pencil, ribbon, edge of book, table, etc.

Plane
It is a flat surface. A plane has length and width, but no thickness.
In the figure, there are six plane surfaces of the cube. Triangles,
quadrilateral, pentagon, etc. are some examples of plane figure.

12.2 Intersecting line segments
Let's study the following illustrations.

a) Two roads meeting at a place making cross roads.

b) Two arms of scissors meet at a point.

c) A The minute hand OA and the hour hand OB of a watch meet each
other at a point O.
O
B

In each of the above figures, the line segments cross each other at a point.

Thus, two or more line segments are said to be the intersecting line segments if
they cross each other through a common point.

In the figure (i), line segments AB and CD
cross each other through a common point O.
So, AB and CD are intersecting line segments.
Similarly, in figure (ii), AB, CD and EF are
intersecting line segments.

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Geometry – Point and Line

12.3 Parallel line segments AP S Q
D R
Let's observe the opposite edges of your
textbook, table, desk, whiteboard, etc.

and investigate idea about the parallel B

lines. C
Two or more line segments are said to

be parallel if the perpendicular distance

between them is always equal. Parallel line segments do

not meet each other when they are produced to either

directions.

In the figure, AB and CD are two parallel line segments
because their perpendicular distances PQ and RS are
equal.

Parallel line segments are represented as and AB
parallel to CD is written as AB//CD.

Note that, if AB // CD and CD // EF then AB // EF.

12.4 Perpendicular line segments 90°
90°
Let's observe angle between each stair, adjacent edge of your
exercise book, angle between the hands at 9 o'clock in a watch, etc. stairs
and investigate the idea of perpendicular lines.

Two line segments are said to be
perpendicular if they meet (or intersect)
each other making an angles of 90q. In
the figure (i), CD is perpendicular to AB
at D. We write it as CD A AB. Similarly,
in the figure (ii) RS A PQ at O.

Note that, if AB A MN and CD A MN
then AB A CD.

EXERCISE 12.1

General Section – Classwork
1. Let's read the incomplete sentences given below and write 'parallel' or

'perpendicular' to complete them.

a) The opposite edges of a ruler are .....................

b) The arrangements of books in the bookshelf are .....................

c) The X-axis and Y-axis are .....................

d) The adjacent sides of a rectangle are .....................

e) The diagonals of a square are .....................

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Geometry – Point and Line

2. Let's tell and write the answers as quickly as possible.

a) How many straight lines can be drawn from a point ? ................

b) How many straight lines can be drawn through two points ? ................

c) How many curved lines can be drawn through two points ? ................

d) Through how many points do two lines interest each other ? ..................

e) What is the angle made by two perpendicular lines ? ..................

f) What is the angle made by two parallel lines ? ..................

g) If the perpendicular distances between two lines at any

point are equal, the lines are said to be ..................

h) Are the opposite edges of your desk parallel ? ..................

i) Are the breadths of your books perpendicular to their lengths ? ...............

j) How many plane surfaces are there in your exercise book ? ..................

3. Look at the adjoining figure. Tell and write the pairs of D C
parallel and perpendicular lines in geometrical notations.
O
Parallel lines are : (Hint : AB// CD) ............................. A PB
Perpendiculars are : ....................................................

4. a) If AB//CD and CD//EF, what is the relation between AB and EF?

........................................................................
b) If PQ A AB and RS A AB, what is the

relation between PQ and RS?

................................................................

c) If KL A PQ and MN A KL, what is the relation
between MN and PQ?

................................................................

d) If AB A XY and CD A AB, what is the relation
between XY and CD?

................................................................

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Geometry – Point and Line

e) If AB // CD, EF and GH are perpendicular lines A E GB
on AB and CD, what is the relation between EF F HD
and GH?

................................................................ C

f) If WX = YZ, what is the relation between AB C
and CD?

................................................................

g) How many perpendiculars can be drawn from
the point P on the line AB?

................................................................

h) How many lines parallel to PQ can be drawn
through the point A?

................................................................

i) If KL is perpendicular to XY at L but it is
not perpendicular to PQ. What do you say
about XY and PQ?

................................................................

Creative Section

5. Name the points that represent the vertices of these figures.

a) A b) S R c) D
C
E

B C Q
P
AB

6. Draw these straight lines, curved lines, rays, line segment in separate

groups. Also write their names. Y NF

P QR S

CD

X

ME

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Geometry – Point and Line

7. Name the line segments and their points of intersection. X
P
a) A C b) R T M c)

K L

O P Q MQ N
A

DB NU S Y

8. Name the parallel and perpendicular line segments. Also represent them in
geometrical notation.

E.g. AB parallel to CD (AB//CD). c) d)
AB perpendicular to CD (AB A CD)
a)
b)

9. Name the parallel and perpendicular line segments. Express them in
geometrical notations.

a) b) c)

10. Copy the tables and write the measurements of the perpendicular distance
between each pair of line segments. State whether the line segments are
parallel or not.

PQ RS TU KL MN OP AB CD EF

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Unit Perimeter, Area and Volume

16

16.1 Perimeter, area and volume - Looking back

Classwork - Exercise

1. Let’s tell and write the area as quickly as possible.

a) l = 3 cm, area of square = .....................

b) l = 8 cm, area of square = .....................

c) l = 4 cm, b = 2cm, area of rectangle = .....................

d) l = 7 cm, b = 5 cm, area of rectangle = .....................

2. Let’s tell and write the perimetres as quickly as possible.

a) l = 3 cm, perimeter of square = ................

b) l = 5 cm, perimeter of square = ................

c) l = 3 cm, b = 2cm, perimeter of rectangle = ................

d) l = 6 cm, b = 4 cm, perimeter of rectangle = ................

3. Let’s tell and write the volumes as quickly as possible.

a) l = 2 cm, volume of cube = ......................

b) l = 3 cm, volume of cube = ......................

c) l = 4 cm, b = 2 cm, h = 1 cm, volume of cuboid = .............
d) l = 5 cm, b = 3 cm, h = 2 cm, volume of cuboid = .............

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Perimeter, Area and Volume

16.2 Length and Distance
1. Let’s discuss on the following questions.

a) Can you guess the length and breadth of your mathematics book?
Let’s measure them by using a ruler.

b) Can you guess the height of your best friend?
Let’s measure it by using a ruler or a measuring tape.

c) Can you guess the length and breadth of your classroom?
Let’s measure them by using a measuring tape.

d) Can you guess the distance between your house and school?

The measurement of farness (or closeness) between two points (or two ends)
is called length. Lengths are measured in millimeter (mm), centimeter (cm),
meter (m), inch (in), foot (ft) etc. The length of the space between two points
(or places) is called distance. We measure distance in meter (m), kilometer
(km) or in mile.

2. Let’s tell and write the answer as quickly as possible.

a) There are ….. millimeters (mm) in 1 centimeter (cm).

b) In 1 meter (m), there are …..centimeters (cm). 1 cm = 10 mm
c) 1 kilometer (km) is equal to ….. meters (m). 1 m = 100 cm
d) The length of 1 meter long stick is …. mm. 1 km = 1000 m
e) The distance of 1 km has …. cm.

Relation between different units of length
Let’s look at the ruler given aside and guess
how many centimeters can make 1 inch.

Let’s take a measuring tape and observe the scales in cm,
inch and foot. Then establish the following relationship
among cm, inch, foot and meter under discussion.

1 inch (in) = 2.54 cm
1 foot (ft) = 12 inch (in)
1 foot (ft) = 12 ×2.54 cm = 30.48 cm

1 meter (m) = 100 cm = 100 inch = 39.37 inch (in)
2.54
39.37
1 meter (m) = 39.37 inch = 12 feet = 3. 28 feet (ft)

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Perimeter, Area and Volume

Workedout examples

Example 1: Convert 6m into cm, in and ft.

Solution:

Here,

To convert 6m in to cm, 1m = 100 cm ? 6m = 6 ×100 cm = 600 cm

To convert 6m in to in, 1m = 39.37 in ? 6m = 6 ×39.37 in = 236.22 in

To convert 6m in to ft, 1m = 3.28 ft ? 6m = 6 ×3.28 ft = 19.68 ft

Example 2: Express each of the following measurements in to cm.
a) 5m 40 cm b) 18 in c) 6 ft d) 5 ft 3 in

Solution:
a) 1m = 100 cm

? 5m 40 cm = 5m + 40 cm

= 5×100 cm + 40 cm = 500 cm + 40 cm = 540 cm

b) 1 in = 2.54 cm ? 18 in = 18×2.54 cm = 45.72 cm

c) 1 ft = 30.48 cm ? 6 ft = 6×30.48 cm = 182.88 cm

d) 1 ft = 30.48 cm and 1 in = 2.54 cm

? 5 ft 3 in = 5 ft + 3 in = 5×30.48 cm + 3 ×2.54 cm

= 152.4 cm + 7.62 cm = 160.02 cm

Example 3: A tree is 25 ft 8 in high. Covert the height of the tree into inch
and centimetre.

Solution:
Here, the height of the tree = 25 ft 8 in
Now, 1ft = 12 in
? The height of the tree = 25 ft 8 in

= 25 ft + 8 in = 25 ×12 in + 8 in
= 300 in + 8 in = 308 in

Also, 1ft = 30.48 cm and 1 in = 2.54 cm
? The height of the tree = 25 ft 8 in = 25 ft + 8 in

= 25 ×30.48 cm + 8 ×2.54 cm
= 762 cm + 20.32 cm =782.32 cm

Example 4: If the length of a piece of carpet is 560 cm, find the length of the
carpet in m, in and ft.

Solution:
Here, the length of carpet = 560 cm

Now, 100 cm = 1 m or, 1 cm = 1 m
100
560
? the length of carpet = 560 cm = 100 m = 5.6 m

3.54 cm = 1 in or, 1 cm = 1 in
3.54

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Perimeter, Area and Volume

? the length of carpet =560 cm = 560 in = 220.47 in
2.54
1
30.48 cm = 1ft or, 1 cm = 30.48 ft

? the length of carpet = 560 cm = 560 ft = 18.37 ft
30.48

Example 5: The length and breadth of a play-ground are 180 feet 6 inch and

120 feet 9 inch respectively. Find the length and breadth of the

ground in m and cm.

Solution:

Here, the length of ground = 180 ft 6 in
6
= 180 ft + 12 ft = 180 ft + 0.5 ft = 180.5 ft

? The length of ground = 180.5 ft = 180.5 m
3.28

= 54.88 m = 54 m + 0.88×100 cm = 54m 88 cm

Again, the breadth of ground = 120 ft 9 in = 120 ft + 9 ft
12

= 120 ft + 0.75 ft = 120.75 ft

? The breadth of ground = 120.75 ft = 120.75 m
3.28

= 36.81 m = 36 m + 0.81×100 cm = 36m 81 cm

Example 6: If the length of a mat is 2m 50 cm and breadth is 4 ft 6 in, by how

many feet is the length of the mat more than the breadth?

Solution: 50
100
Here, the length of the mat = 2m 50 cm =2m + m = 2m + 0.5 m= 2.5 m

? The length of the mat in feet = 2.5×3.28 ft = 8.2 ft.

Again, the breadth of the mat = 4 ft 6 in = 4ft + 6 ft = 4 ft + 0.5 ft =4.5 ft
12

Difference of length and breadth of the mat = l – b = 8.2 ft – 4.5 ft = 3.7 ft

Hence, the length of the mat is 3.7 ft more than its breadth.

Example 7: Sunayana bought 4 ft 9 in long ribbon and her friend Anita bought

3 ft 6 in long ribbon in a shop. How many meters of ribbon did

they buy altogether?

Solution: 9
12
Here, the length of ribbon for Sunayana = 4 ft 9 in =4 ft + ft

= 4 ft + 0.75 ft= 4.75 ft
1.75
?The length of the ribbon for Sunayana = 3.28 m = 1.45 m

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Perimeter, Area and Volume

Again, the length of ribbon for Anita = 3 ft 6 in =3 ft + 6 ft
12

= 3 ft + 0.5 ft= 3.5 ft

? The length of the ribbon for Anita = 3.5 m = 1.07 m
3.28

Total length of the ribbon = 1.45 m + 1.07 m = 2.52 m

Hence, they bought 2.52 m long ribbon altogether.

EXERCISE 16.1
General Section - Classwork

1. Let’s tell write the correct answers in the blank spaces.

a) There are ….. cm in 1 m.

b) 1 inch equals to … cm.

c) The length of 1 ft long wire is …. inch.

d) A 1 ft long ruler has …. cm.

e) The length of 1 m long stick is … inch.

2. Let’s choose and tick the correct option in the following questions.

a) How many centimeters are there in 1 inch?

(i) 2.54 (ii) 3.54 (iii) 12 (iv) 30.48

b) How many inches are there in 1 foot?

(i) 3.28 (ii) 12 (iii) 30.48 (iv) 39.37

c) 1 meter is equal to ….

(i) 3.28 ft (ii) 2.54 ft (iii) 30.48 ft (iv) 39.37 ft

d) Which of the following measurement is the shortest?

(i) 1 cm (ii) 1 in (iii) 1 ft (iv) 1 m

e) If he distance between your house and school is 200 m. The distance
of the school from house measured in feet is …

(i) 508 (ii) 656 (iii) 6,172 (iv) 7,874

Creative Section-A
1. Convert each of the following measurements in to cm.

a) 5 m b) 18 m c) 20 m 25 cm d) 37 m 40 cm e) 6 ft

f) 14 ft g) 6 in h) 10 in i) 5 ft 3 in j) 8 ft 9 in

2. Change each of the following measurements in to inch.

a) 4 m b) 20 m c) 16 m 50 cm d) 48 m 25 cm e) 7 ft

f) 24 ft g) 35.5 ft h) 48. 75 ft i) 25 ft 7 in j) 53 ft 11 in

3. Change the following measurements in to ft.

a) 15 m b) 28 m c) 40 m 20 cm d) 55 m 55 cm e) 381 cm

f) 76.2 cm g) 24 in h) 66 in i) 7 ft 6 in j) 28 ft 3 in

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4. Express the following measurements in to m.

a) 12 m 60 cm b) 35 m 25 cm c) 32.8 ft d) 53.3 ft
e) 3937 in f) 727.4 in g) 25 ft 6 in h) 36 ft 9 in
Creative Section-B

5. a) Bishwant is 5 feet 6 inch tall. Find his height in inch, centimeter and
meter.

b) The length of a kitchen garden is 10 m 50 cm and breadth is 6 m 20 cm.
Find the length and breadth of the kitchen garden in ft.

c) The height of Mt. Everest is 8,848 m 86 cm and the height of
Kanchenjunga is 8,586 m, find their heights in feet.

d) The length and breadth of a school’s playing ground are 220 feet 9
inch and 150 feet 3 inch respectively. Find the length and
breadth of the ground in m.

6. a) Ram is 137.16 cm tall and Sita is 4 ft 3 in tall. Who is taller and by how
many feet? Find.

b) If the length of whiteboard of class-VI is 2m 80 cm and breadth is
4 ft 9 in, by how many meters is the length of the whiteboard more than
the breadth? Find.

c) For school’s uniform, Shashwat needs 1 m 50 cm cloth for a shirt and
5 ft 8 inch cloth for a pant. How long cloth does he require for the
uniform? Find in meter.

d) In a village, there are three vertical electric poles. If the distance between
the first and the second poles is 98 ft 3 in and the distance between the
second and the third poles is 48 m 60 cm. Estimate the shortest length
of electrical cable wire that joins these three poles in ft.

It’s your time- Project work!

7. a) Let’s make a group of your 5 friends and measure the height of each in
ft using a measuring tape. Convert the height of each member of your
group in cm, in and m. Then compare the heights converted in cm, in
and m and the corresponding values in measuring tape.

b) Let’s measure the width (left to right) and length (top to bottom) of the
door of your bedroom in ft. Then convert them in to meter.

c) Let’s measure the length, breadth and height of your classroom in
meter. Then convert them in ft and inch.

16.3 Perimeter of plane figures

The total length of the boundary line of a plane figure is called its perimeter.
(i) Perimeter of triangles

Triangle ABC is bounded by 3 sides. So, the total
length of the boundary is AB + BC + CA

= (c + a + b) cm
= (a + b + c) cm

? Perimeter of a triangle = a + b + c

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(ii) Perimeter of rectangles

The opposite sides of a rectangle are equal.
So, the lengths AB = DC = l

the breadth BC = AD = b

The perimeter of the rectangle ABCD = AB + BC + CD + DA
= l + b + l + b = 2l + 2b = 2 (l + b)

? Perimeter of a rectangle = 2(l + b)

(iii) Perimeter of regular polygons
The length of each side of a regular polygon is equal. study the table given
below and learn to find the perimeter of regular polygons.

Regular polygon Number of sides Length of a side Perimeter

Equilateral 3 l l + l + l = 3l
triangle

Square 4 l l + l + l + l = 4l

Regular 5 l l + l + l + l + l = 5l
pentagon

Regular 6 l l+l+l+l+l+l
hexagon = 6l

Regular n l l + l + l + … n times
polygon = nl

(iv) Perimeter of an isosceles triangle A a
Two sides of an isosceles triangle are equal. a

So, equal sides AB = AC = a

and base BC = b Bb C

The perimeter of the isosceles triangle ABC = AB + BC + AC

=a+b+a

= 2a + b

Thus, perimeter of an isosceles triangle = 2a + b

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Worked-out examples

Example 1: Find the perimeter of the following figures.

a) 1 cm 1 cm b)

3 cm 2 cm

4 cm

Solution:

a) 1 cm 1 cm ∴Perimeter= 4 cm + 3 cm + 1 cm + 2 cm +
2 cm 2 cm + 2 cm + 1 cm + 3 cm
3 cm 2 cm = 18 cm
2 cm 3 cm

4 cm

b)

Perimeter = 10 cm + 2 cm + 3 cm + 2 cm +
3 cm + 2 cm + 4 cm + 6 cm

= 32 cm

Example 2: If the perimeter of an equilateral is 24 cm, find the length of its

side.

Solution:

Here, perimeter of the equilateral triangle = 24 cm

or, 3l = 24 cm
24
or, l = 3 = 8 cm

Hence, the required length of the side of the triangle is 8 cm.

Example 3: The perimeter of an isosceles triangle is 18.5 cm. If the length of

one of its equal sides is 4.8 cm. Find the length of remaining side.

Solution:

Here, perimeter of the isosceles triangle = 18.5 cm

length of equal sides (a) = 4.8 cm

remaining side (b) = ?

Now, perimeter (p) = 2a + b

or, 18.5 cm = 2 × 4.8 cm + b

or, 18.5 cm = 9.6 cm + b

or, 18.5 cm – 9.6 cm = b

b = 8.9 cm

∴ Remaining side of the triangle is 8.9 cm.

Example 4: A rectangular ground is 15 m long and 10 m broad. Find the length
of a wire required to fence it with 3 rounds.

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Solution:

Here, length of the ground (l) = 15 m I understand!
breadth of the ground (b) = 10 m 1 round fencing is equal to
it’s perimeter.
Now, perimeter of the rectangular ground So, 3 round fencing
= 3 × perimeter
= 2 (l + b)
= 2 (15 + 10) m
= 2 u 25 m = 50 m.

? The required length of the wire = 3 u 50 m = 150 m

EXERCISE 16.2
General Section – Classwork

1. Let’s tell and write the perimetres of these figures as quickly as possible.

a) b) 3 cm c) 7 cm
3 cm
3 cm 4 cm 3 cm 3 cm

3 cm

5 cm 7 cm

3 cm

perimeter = ............. perimeter = ............. perimeter = .............

2. Let’s tell and write the answers as quickly as possible.

a) If the sides of a triangle are x cm, y cm and z cm.

Its perimeter = .......................

b) If each of the sides of a square is l cm.

Its perimeter = .......................

c) If the length and breadth of a rectangle are l cm and b cm respectively.

Its perimeter = .......................

d) If the length of each side of a polygon is l cm and the number of sides

is n, the perimeter of the polygon = .......................

e) If the length of each side of a regular octagon is l. It’s perimeter = ...........

Creative Section - A

3. Find the perimeters of these figures.

a) b) c) d)

4 cm

4 cm 5 cm

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e) f)

4. a) The length of a side of equilateral triangle is given below. Find the
perimetres of triangles.

(i) l = 5 cm (ii) l = 4.5 cm (iii) l = 2.6 cm

b) The length of a side of square is given below. Find the perimetres.

(i) l = 4 cm (ii) l = 6.5 cm (iii) l = 7.2 cm

c) The length and breadth of rectangle are given below. Find the
perimeters.

(i) l = 5. 5 cm, b = 4 cm (ii) l = 6.3 cm, b = 4.7 cm (iii) l = 7.8 cm, b = 4.7 cm

d) The length of each side of regular polygons is given below. Find the
perimeters.

(i) pentagon l = 3 cm (ii) hexagon l = 4.5 cm (iii) octagon l = 6.5 cm

5. a) The perimeter of an equilateral triangle is 18 cm, find the length of its
each side.

b) If the perimeter of an equilateral triangle is 45 cm, what is the length of
its each side?

6. a) The perimeter of an isosceles triangle is 32 cm and the length of each
equal side is 10 cm. Find the length of its base.

b) In 'ABC, the length of sides AB and AC is 6 cm each. The perimeter of
'ABC is 20 cm. Which is the longest side of the triangle?

7. a) A squared field is 20 m long. If you are running around it, how many
metres do you travel in one round ?

b) A rectangular garden is 35 m long and 25 m broad. How many metres
does a girl cover in one complete round around it ?

8. a) If the perimeter of a squared ground is 220m, find the length of a wire
required to fence with

(i) 1 round (ii) 2 rounds (iii) 3 rounds

b) The perimeter of a rectangular field is 180m. Find the length of a wire
required to fence it with 4 rounds.

9. a) A squared park is 80 m long. How many metres of wire is required to
fence it with 5 rounds?

b) You are running around a rectangular ground of length 42 m and breadth
24 m. How many metres do you cover when you complete 6 rounds
around it?

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10. a) If the perimeter of a square is 84 cm, find its length.

b) The perimeter of a rectangle is 56 cm and its length is 18 cm. Find its
breadth.

c) A rectangular compound is 32 m broad and its perimeter is 144 m. Find
its length.

Creative Section -B
11. a) Your friend Anuradha has drawn a triangle whose length of sides are in

the ratio of 2:3:4 and perimeter is 18 cm, find:

(i) the length of each side.

(ii) difference between the longest and shortest sides.

b) The ratio of sides of a triangular garden is 4:5:2. If the perimeter of the
garden is 110 m, find:

(i) the length of each side.

(ii) by how much is the longest side more than its shortest side?
1cm 1cm
12. a) Once in a classroom, teacher gave 4 square
tiles to Samriddhi and Surav each. Then 1cm

ask them to arrange them in rectangular or 1cm
square form. Samriddhi arranged the tiles

in square form and Saurav arranged in the

rectangular form, whose perimeter is more 1cm

and by how much? 1cm 1cm 1cm 1cm

b) Find the perimeter of each figure formed by using 6 square tiles. By what

percentage is the maximum perimeter more than the minimum one?

13. a) A wire is in the shape of rectangle. Its length is 6 cm and breadth is 4 cm.
If the same wire is rebent in the shape of square, what will be the length
of each side?

b) A rope is in the square shape. It’s each side is 6 cm. If the same wire
is rebent in the rectangular shape with length 8 cm, what would be its
breadth?

It’s your time - Project!
14. a) Let’s measure the length and breadth of the surfaces of desks (or tables) inside

your classroom. Find the perimeter of each surface and compare them.
b) Let’s measure the length and breadth of the floor of your classroom and

calculate the perimeter of the floor.
c) Let’s measure the length and breadth of your school playground. Estimate

the cost of fencing it with 5 rounds using metal wires at the local rate of
cost of wire per meter.

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16.4 Area of plane figures
Area of a plane figure is the plane surface enclosed by the boundary
line of the figure.
The length of each side of the adjoining square is 1 cm.
So, the surface enclosed by this square is 1 square cm (or 1 cm2).
? The area of this square = 1 cm2

(i) Area of rectangles
Let the area of each squared room of the graph given alongside is 1 cm2. So,
the surface enclosed by the given rectangle is 8 cm2.
? Area of the rectangle is 8 cm2.
Here, along the length it encloses 4 squared rooms.
Along the breadth it encloses 2 squared rooms.
It’s area = 8 cm2 = (4 × 2) cm2 = length u breadth
Thus, area of a rectangle = length u breadth = l u b

(ii) Area of squares
The length of each side of a square is equal.
So, in a square, length = breadth = l
As like a rectangle, area of the square = length u breadth = l u l = l2
Thus, area of a square = l2

Worked-out examples

Example 1: From the given graph, find the area b

of the figures.

Solution:

a) Here, the number of complete squared rooms

= 16

The number of half squared rooms = 4 = 2

complete squared rooms.

? The area of the figure = (16 + 2) cm2 = 18 cm2.

b) Here, the number of complete squared rooms = 15

The number of squared rooms which are more than half size = 12

Neglecting the squared rooms less than half size and counting the squared

rooms more than half size as 1,

The approximate area of the shape = (15 + 12) cm2 = 27 cm2.

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Example 2 : Find the area of these figures. b) 10 cm
a) 8 cm
A
X 5 cm B 3 cm 6 cm
C
3 cm Y

2 cm 2 cm 2 cm

Solutions :

a) Area of X - part = 8 cm × 5 cm = 40 cm2

Area of Y - part = 3 cm × 2 cm = 6 cm2

So, the area of the figure = 40 cm2 + 6 cm2 = 46 cm2.

b) Area of A - part = 10 cm × 3 cm = 30 cm2

Area of B - part = 2 cm × 3 cm = 6 cm2

Area of C - part = 2 cm × 3 cm = 6 cm2

Hence, the area of the figure = 30 cm2 + 6 cm2 + 6 cm2 = 42 cm2

Example 3 : The area of a rectangle is 54 cm2 and it is 9 cm long. Find its
breadth.

Solution :

Here, area of the rectangle = 54 cm2

or, l × b = 54 cm2

or, 9 cm × b = 54 cm2

or, b = 54 cm2 = 6 cm
9 cm

Hence, the required breadth of the rectangle is 6 cm.

Example 4: The perimeter of a rectangular garden is 70 m and its length is 20 m.
a) Find its breadth b) Find its area.

Solution :
a) Here, perimeter of rectangle = 70 m

or, 2 (l + b) = 70 m

or, 2 (20m + b) = 70 m

or, 40 m + 2b = 70 m

or, 2b = 70 m – 40 m = 30 m

or, b = 15 m

Hence, the required breadth is 15 m.

b) Now, area of the field = 20 m × 15 m = 300 m2.
Hence, the required area of the field is 300 m2.

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Example 5: The area of a rectangular floor is 50 m2. If its length is two times

of its breadth,

a) find its length and breadth b) find its perimeter.

Solution:

a) Let the breadth of the floor be x m.

According to the question, its length = 2x cm

Now, area of the rectangular floor = 50 m2

or, l u b = 50 m2

or, 2x u x = 50 m2

or, 2x2 = 50m2
50
or, x2 = 2 m2 = 25 m2

or, x = 25 m2

=5m

So, breadth = x = 5 m and length = 2x = 2 u 5 m = 10 m

b) Again, the perimeter of the rectangular floor = 2(l + b)
= 2(10 m + 5 m)
= 2 × 15 m = 30 m

Example 6: Calculate the area of the shaded region in 4 cm 7 cm
the figure alongside. 6 cm

Solution:

10 cm

Area of the bigger rectangle = 10 cm u 7 cm = 70 cm2
Area of the smaller rectangle = 6 cm u 4 cm = 24 cm2
Now, area of the shaded region is
Area of the bigger rectangle – Area of the smaller rectangle = 70 cm2 – 24 cm2

= 46 cm2

EXERCISE 16.3

General Section – Classwork

1. Let’s tell and write the area of these figures AB
from the graph. C

a) Area of figure A = ...................

b) Area of figure B = ...................

c) Area of figure C = ...................

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2. Let’s tell and write the answer as quickly as possible.

a) The plane surface enclosed by the boundary lines of a figure is called

................ of the plane figure.

b) The total length of boundary lines of a plane figure is its ................

c) Each sides of a square is x cm. Its area is ................ cm2.

d) The length and breadth of a rectangle are a cm and b cm respectively.

Its area is ................

3. Let’s tell and write the area as quickly as possible.

a) l = 5 cm, b = 3 cm, area of rectangle = ..................

b) l = 6.5 cm, b = 2 cm, area of rectangle = ..................

c) l = 7 cm, area of square = ..................

d) l = 8 cm, area of square = ..................

e) Perimeter of a square = 20 cm, its area = ..................

f) Perimeter of a square = 40 cm, its area = ..................

4. Let’s tell and complete the table as quickly as possible.

a) Length (l) Breadth (b) Area (A)
Rectangle 5 cm 3 cm ........................
A

B 8 cm 4.5 cm

C 10 cm .................. 60 cm2

D 9 cm .................. 54 cm2

E .................. 4 cm 28 cm2

F .................. 7 cm 63 cm2

b) Length (l) Perimeter (P) Area (A)
Square 5 cm
A .................. ........................

B 8 cm .................. ........................

C .................. 12 cm ........................

D .................. 20 cm ........................

E .................. .................. 25 cm2

F .................. .................. 100 cm2

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Creative Section -A
5. Find the area of these figures from the graph given below.

6. Find the area of the following figures. 4.5 cm

7 cm

7. Find the area of the rectangles whose lengths and breadths are given below.

a) l = 6 cm, b = 4 cm b) l = 9.5 cm, b = 8 cm

c) l = 12.5 cm, b = 9.4 cm d) l = 15.2 cm, b = 10.8 cm

8. Find the area of the squares whose sides are given below.

a) l = 6 cm b) l = 5.5 cm c) l = 10.5 cm d) l = 16.2 cm

9. a) A square room is 5 m long. Find the area of carpet required to cover its
floor. Hint : Area of carpet = Area of the floor

b) A rectangular room is 6 m long and 5.5 m broad. Find the area of carpet
required to cover its floor.

c) A rectangular hall is 10 m long and 8.5 m broad.

(i) Find the area of carpet required to cover its floor.

(ii) If the cost of 1 m2 of carpet is Rs 75, find the cost of carpeting the
floor.

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10. a) The area of a square is 49 cm2.

(i) Find its length (ii) Find its perimeter

b) If the area of a squared garden is 196 m2, find its perimeter.

c) The area of a rectangle is 70 cm2 and it is 7 m broad.

(i) Find its length (ii) Find its perimeter

d) The area of a rectangular surface of a table is 6 m2 and it is 3 m long.
Find its perimeter.

11. a) The area of a rectangle is 18 cm2. It’s length is double than that of its
breadth.

(i) Find its length and breadth (ii) Find its perimeter

b) The area of a rectangular floor is 32 m2. If its breadth is half of its
length, find its perimeter.

12. a) The perimeter of a square is 12 cm.

(i) Find its length (ii) Find its area

b) The perimeter of a squared field is 60 m. Find its area.

c) The perimeter of a rectangle is 28 cm and its length is 8 cm.

(i) Find its breadth (ii) Find its area

d) The perimeter of a rectangular floor is 40 m and it is 8 m broad. Find its
area.

13. Find the area of the following shaded regions.

Creative Section -B

14. a) The students of a school with two banners of equal area were participating
in a rally on ‘Children’s Day’. The first banner was 8 ft. long and 3 ft.
wide. If the second banner was 6 ft. long, what was its width?

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b) Mr. Khadka has bought two pieces of carpet for his room. Carpet-X has
area 48 sq. ft and width 6 ft. The area of carpet - Y is one-half area of
carpet-X. If both the carpets have same length, what is the width of
carpet-Y?

15. a) The area of a squared lawn and rectangular vegetable garden are equal.
The perimeter of the lawn is 24 m and its side is twice the breadth of the
garden. Find the length of the garden.

b) The area of a rectangular playground and a squared park are equal. The
perimeter of the park is 80 m and its side is half of the length of the
ground. Find the width of the ground.

16. The following figures have equal perimeters.

(i) Do they have equal
areas?

(ii) Which one has the 6 cm 9 cm
more area and by
how much? Explain. 12 cm 9 cm

It’s your time- Project work!

17. a) Let’s measure the length and breadth of the surface of your maths book
and find its area.

b) Let’s measure the sides of the surface of your desk (or table) and find it’s
area.

16.5 Introduction of solid figures - review
The table given below shows some solid figures.

Solid figures Name Examples

Cuboid

Book Butter

Sphere

Ball Globe

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Cylinder

Pencil Water jar

Cone

Ice–cream Traffic-divider

Pyramid

Tent

16.6 Faces, edges and vertices of solid figures

Let’s look at the adjoining cuboid. The rectangular
surface of the cuboid are called its faces. The joint of
two rectangular faces are called its edges. The corners at
which three edges meet each other are called vertices.

Let’s count the number faces, edges and vertices of the following solids and
complete the table.

Cuboid Triangular prism Pyramid

Solid figures Number of faces Number of Number of edges F+V–E
F vertices E
6 + 8 – 12 = 2
Cuboid 6 V 12 ………………..
………………..
Triangular ……………….. 8
………………..

prism

Pyramid ……………….. ……………….. ……………….. ………………..

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Does the relation F + V – E = 2 hold true in the cases of triangular prism and
pyramid?

This rule was developed by Swiss Mathematician Euler. So, it is also called
Euler’s rule. The solids which hold the Euler’s rule true are also called Polyhedra.
So, the rule is very much useful to test whether a solid is polyhedron or not.

16.7 Construction of some models of solids

We can make the models of solid figures by folding paper. To make such models
we should first draw their nets on the papers. Such skeletal models can also be
made by using match–sticks, pieces of straws, etc.

The table given below of the solid figures shows their nets and skeleton models.

Name of Solids Geometrical figure Net Skeleton

Cube

Tetrahedron

Pyramid

Octahedron

Draw the nets of the above solid figures on the separate hard–papers. Cut the
nets out and fold along the dotted lines. Paste the edges of the folded faces with
glue. Now, you have prepared the models of the solid figures.

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EXERCISE 16.4
General Section – Classwork
1. Let’s tell and write the answers as quickly as possible.

a) Number of faces of a cube are ......................

b) Number of edges of a cuboid are ......................

c) Number of vertices of a cuboid are ......................

d) Number of faces of a cylinder are ......................

e) Number of vertices of a pyramid are ......................
2. Let’s tell the name of these solid figures and write in the blanks.

a) b) c) d)

...................... ...................... ...................... ......................

3. Look at the nets and identity these solid figures.

a) b) c) d)

...................... ...................... ...................... ......................

Creative Section - A

4. Give any two examples of the following solids.

a) Cube b) Cuboid c) sphere d) cylinder e) cone

5. Draw the following solid figures. Write the number of their face (F),
vertices (V) and edges (E) and so that F + V – E = 2 in each case.

a) Cuboid b) Tetrahedron c) Triangular prism d) Pyramid

6. Draw the following nets on separate hard papers. Cut the outlines of the nets
out and fold along the dotted lines. Paste the edge of the folded faces with
glue. Name the solids you have made.

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7. a) Take 6 match–sticks and make the skeleton model of a tetrahedron.
b) Take 8 match–sticks and make the skeleton model of a pyramid.
c) Take 12 match–sticks and make the skeleton model of a cube.
d) Take 12 match–sticks and make the skeleton model of an octahedron.

16.8 Area of solids
In this class, we shall find the area of cuboid and cube.

(i) Area of cuboid

The solid figure given alongside is a cuboid. It has 6
rectangular faces. It’s area is the total sum of the area of
6 rectangular faces.

Area of the top and bottom faces = lb + lb = 2lb

Area of the side faces = bh + bh = 2 bh

Area of the front and back faces = lh + lh = 2lh
? Area of cubiod = 2 lb + 2 bh + 2lh = 2 (lb + bh + lh)

(ii) Area of cube
A cube has 6 squared faces. Each squared face has the area of l2.
? Area of cube = l2 + l2 + l2 + l2 + l2 + l2 = 6l2

16.9 Volume of solids
The volume of a solid is defined as the total space occupied by itself. Volume
is measured in cubic millimetre (cu. mm or mm3), cubic centimetre (cu. cm or
cm3), cubic metre (cu. m or m3), etc.

The solid given alongside is a cube. It’s length, breadth and
height are of 1 cm each. It’s volume is said to be 1 cm3.

(i) Volume of cuboid
The volume of a cuboid is calculated as the
product of its length, breadth and height.

? Volume of a cuboid = length u breadth u height
=lubuh

(ii) Volume of cube:
The length, breadth and height of a cube are equal.
i.e. l = b = h
? Volume of a cube = l u b u h = l u l u l = l3

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Perimeter, Area and Volume

Worked-out examples

Example 1: A rectangular block is 5 cm long, 4 cm broad and 3 cm high. Find
its a) total surface area b) volume

Solution:
a) Here length of the block (l) = 5 cm

breadth of the block (b) = 4 cm
height of the block (h) = 3 cm

Now, the total surface area of the block = 2 (lb + bh + lh)
= 2 (5 u 4 + 4 u 3 + 5 u 3) cm2
= 2 (20 + 12 + 15) cm2 = 2 u 47 cm2
= 94 cm2

b) Again, the volume of the block = l u b u h
= 5 cm u 4 cm u 3 cm
= 60 cm3

Example 2: If the total surface area of a cube is 54 cm2, find its volume.

Solution:

Here, the total surface area of the cube = 54 cm2

or, 6l2 = 54 cm2

or, l2 = 54 cm2
6

or, = 9 cm2

? l = 9 cm2 = 3 cm.
Now, volume of the cube = l3

= (3 cm)3

= 27 cm3

Example 3: The length of a cuboid is two times of its breadth and its height is
3 cm. If the volume of the cuboid is 24 cm3, find its length and
breadth.

Solution:
Let the breadth (b) of the cuboid be x cm.

So, its length (l) will be 2x cm.

The height of the cuboid (h) = 3 cm

Volume of the cuboid= 24 cm3

or, l u b u h = 24 cm3

or, 2x u x u 3 cm = 24 cm3

or, 6x2 cm = 24 cm3

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Perimeter, Area and Volume

or, x2 = 24 cm3
6 cm

or, x2 = 4 cm2

or, x = 4 cm2

? x = 2 cm
So, the required breadth (b) = x = 2 cm

and length (l) = 2x
= 2 u 2 cm = 4 cm.

EXERCISE 16.5
General Section – Classwork
1. Let’s tell and write the volume of these solid as quickly as possible.

4 cm 2 cm 3 cm3 cm 10 cm
3 cm
3 cm Volume = .............
Volume = .............
Volume = .............

2. a) l = 2 cm, volume of cube = ................, surface area of cube = ...............

b) l = 3 cm, volume of cube = ..............., surface area of cube = ...............

c) l = 5 cm, b = 4 cm, h = 2 cm, volume of cuboid = ................

d) l = 10 cm, b = 5 cm, h = 4 cm, volume of cuboid = ...............

Creative Section - A
3. Find the total surface area and volume of the cuboids.

a) l = 5 cm, b = 4 cm, h = 3 cm
b) l = 8 cm, b = 5 cm, h = 4.5 cm
c) l = 10.5 cm, b = 4 cm, h = 6 cm,
d) l = 15.8 cm, b = 12.5 cm, h = 10 cm

4. Find the total surface area and volume of the cubes.

a) l = 2 cm b) l = 5 cm c) l = 4.5 cm d) l = 7.6 cm

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Perimeter, Area and Volume

5. a) A rectangular block is 16 cm long, 12 cm broad and 5 cm high. Find its
total surface area and volume.

b) A cubical block is 9 cm long. Find its total surface area and volume.

c) A rectangular tank is 2 m long, 1.5 m broad and 1 m high.
(i) What is the volume of the tank ?
(ii) If it is completely filled with water, what is the volume of water ?
(ii) If 1 m3 = 1000 litres, how many litres of water does it hold ?

d) A rectangular tank is 80 cm long, 60 cm broad and 50 cm high.

(i) What is the volume of the tank ?

(ii) What is the volume of water when it is completely filled ?

(iii) If 1000 cm3 = 1 litre, how many litres of water does it hold ?

6. a) If the total surface area of a cube is 24 cm2,

(i) find its length (ii) find its volume.

b) If the volume of a cube is 125 cm3,

(i) find its length (ii) Find its total surface area

c) If the total surface area of a cube is 96 cm2, find its volume.

d) If the volume of a cube is 27 cm3, find its total surface area.

7. a) The length of a cuboid is two times its breadth and its height is 2 cm. If the
volume of the cuboid is 36 cm3, find its length and breadth.

b) The breadth of a rectangular block is half of its length and its height is
4 cm. If the volume of the block is 200 cm3. Find its

(i) length and breadth (ii) total surface area

Creative Section - B
8. a) The ratio of length, breadth and height of a cuboid is 5:3:2. If its volume

is 240 cm3, calculate its total surface area.

b) A rectangular water tank is 3 m long, 2 m wide and 1.5 m high. How
many litres of water can it hold? [Hint: 1 m3 = 1000 l]

It’s your time - Project work!
9. a) Let’s measure the length, breadth and thickness of your math book. Then

find it’s total surface area and volume.

b) Let’s measure the length, breadth and thickness of the material on the top
of your desk. Then calculate the volume of the material.

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Unit Symmetrical Figures, Design of
Polygons and Tessellations
17

17.1 Symmetrical figures

Let’s look at the following figures. Each of them is divided into two equal parts
and both parts are exactly the same in shape and size.

Such figures which can be divided into two halves and each half is identical to
each other are called symmetrical figures. The dotted line (or lines) that divides
the symmetrical figure into two halves is called axis (or line) of symmetry.

A symmetrical figure may have one, two or more axis (or axes) of symmetry.
Look at the following examples.

It has only one axis of symmetry It has two axes of symmetry It has three axes of symmetry

A isosceles triangle has A rectangle has two An equilateral triangle
only one axis of symmetry axes of symmetry has three axes of symmetry

A square has four A rhombus has two A circle has many
axes of symmetry axes of symmetry (infinite) axes of symmetry

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EXERCISE 17.1

General Section – Classwork

1. Let’s draw as quickly as possible, the axes of symmetry as many possible
of these figures using ruler.

2. Let’s complete the shape of the following figures whose axes of
symmetry are the dotted lines.

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Symmetrical Figures, Design of Polygons and Tessellations

3. Let’s tell and write the possible number of axes of symmetry of these figures.

Ca) The letter .................... Yb) The letter ...................

Pc) The letter .................... d) An isosceles triangle ...................

e) An equilateral triangle ......... f) A circle ......................

Creative Section

4. a) Draw any two figures with a horizontal axis of symmetry.
d) Draw any two figures with a vertical axis of symmetry.
c) Draw any two figures with two axes of symmetry.
d) Draw any two figures with more than two axes of symmetry.

17.2 Design of polygons

We can make attractive designs or patterns by drawing many smaller polygons
inside a bigger polygons or by dividing a bigger polygons into many smaller
polygons and then colouring them with different colours. You can make such
coloured patterns and decorate your classroom or house.

(i) Patterns from square

This pattern is made To make this pattern, at first To make this pattern,

by joining the mid– join the mid–points of every draw as many

points of every side of side of a bigger square to form smaller squares as

a bigger square to make a smaller square. Then, draw possible inside a

a smaller square. the diagonals of the bigger and bigger square.

the smaller squares. In this way

a square is divided into many

small right–angled isosceles

triangle.

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Symmetrical Figures, Design of Polygons and Tessellations

(ii) Patterns from pentagon

To make this pattern, draw as many To make this pattern, draw 5

smaller pentagons as possible inside diagonals of a pentagon to form a star

a bigger pentagon. shape and a small pentagon at the

middle. Again, draw the diagonals of

the smaller pentagon to form a small

star inside it.

(iii) Patterns from hexagon

To make this pattern, draw as many To make this pattern, join every

smaller hexagons as possible inside a two vertices to divide hexagon into

bigger hexagon. triangles and a smaller hexagon is

formed at the middle. Repeat the

same process in the case of smaller

hexagon.

17.3 Tessellation

A tessellation is a covering of the plane with congruent geometrical shapes in
a repeating pattern without leaving any gaps and without overlapping each
other. In a tessellation, often the shapes are polygons. The polygon is either
an equilateral triangle, a square or a regular hexagon. Tessellation is done on
the surface of carpets and surface of floor or wall to make the surface more
attractive.

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Symmetrical Figures, Design of Polygons and Tessellations

Given below are a few examples of tessellations.

Remember, to make a tessellation
(i) Use sets of congruent geometrical figures (Ones that have the same shape

and size)
(ii) Don’t leave any gaps
(iii)Don’t have any overlaps.

EXERCISE 17.2
1. Let’s draw the following patterns similarly and shade them with different

colours.

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Symmetrical Figures, Design of Polygons and Tessellations

2. Let’s study the following pattern. Copy and complete the figure.

3. Let’s draw graphs of your own and copy and complete the following
tessellations.

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Unit Statistics

18

18.1 Collection of data

The heights (in cm) of 20 students of class 6 are given below.

100, 110, 120, 90, 105, 140, 105, 110, 100, 90
120, 95, 100, 120, 130, 110, 140, 125, 115, 110
Such numerical figures are called data. So, data are the information collected
about a particular fact. They are usually in the form of numbers.
Statistics is the branch of mathematics concerned with the collection,
presentation and analysis of data.

18.2 Presentation of data

The above data are not presented in a proper order. So, it is some how difficult
to interpret the performance of the students. Such data are called raw data. How
ever, if the data are presented in a proper order, it will be more convenient to get
the required information for which they are collected. Here, a proper order of
data means they are arranged either in ascending or in descending order.

90, 90, 95, 100, 100, 100, 105, 105, 110, 110
110, 110, 115, 120, 120, 120, 125, 130, 140, 140
Such data in a proper order is called arrayed data.

The proper order of data can be presented by table, bar graphs, pie charts line
graphs etc.

18.3 Frequency table
The data given below represent the weights in kilograms of 15 children of a
class.

20, 25, 30, 25, 35, 25, 20, 35, 25, 35, 35, 40, 25, 20, 30

Here,20 kg is repeated three times. So, the frequency of 20 is 3.
25 kg is repeated five times. So, the frequency of 25 is 5.
30 kg is repeated two times, so, the frequency of 30 is 2.
35 kg is repeated four time. So, the frequency of 35 is 4.
40 kg is repeated only one time. So, the frequency of 40 is 1.

Thus, the number of times the observation occurs or appears in the data is called
the frequency (f) of the observation.

Now, let’s present these data and their frequencies in a table. Such table is called
a frequency table.

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Statistics

Weight (in kg) Tally marks Frequency

20 3

25 5

30 2

35 4

40 1

In a frequency table, we may make a column to show the frequencies by using
tally marks ( ). When the frequency of a data is 5, a tally bar is marked against
four tally bars ( ). In the above table the frequency of 25 kg is 5. So, it is marked
as ( ).

EXERCISE 18.1

General Section

1. Define: (i) data (ii) frequency of an observation.

2. The marks obtained by 20 students of class 6 out of 25 full marks in a class
test are given below. Prepare a frequency table with tally marks and present
the data.
20, 15, 22, 24, 19, 15, 20, 24, 19, 24

22, 19, 20, 24, 15, 20, 24, 19, 20, 22

3. Heights (in cm) of 30 students of a class are given below. Prepare a frequency
table with tally marks and present the data.
110, 100, 95, 100, 120, 115, 110, 125, 95, 110,

120, 125, 110, 115, 120, 100, 130, 110, 120, 115,

110, 115, 125, 110, 115, 100, 110, 120, 115, 110

4. Daily wages (in Rs) of 40 workers in a factory are given below. Present the
data in a frequency table by using tally marks.

75, 60, 80, 55, 90, 85, 70, 65, 80, 75,
80, 65, 70, 80, 55, 90, 80, 60, 70, 95,
90, 75, 65, 70, 75, 80, 95, 80, 80, 70,
85, 75, 80, 60, 65, 70, 80, 85, 80, 85

Creative Section
5. In an interview of 20 married couples about their desired number of

children the response were as follow:
2, 1, 3, 2, 2, 1, 2, 3, 4, 3

2, 1, 1, 2, 1, 2, 3, 2, 1, 2

Present the above data in a frequency table by using tally marks and answer
the following questions.

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Statistics

a) How many couples desired only one child?
b) How many couples desired two children?
c) How many couples desired three children?
d) How many couples desired more than three children?
e) How many couples desired less than 3 children?
f) What is the desirable number of children for the maximum number of

couples?

18.4 Bar graph
Bar graph is one of the simple forms of diagrams representing data along with
their frequencies. In a bar diagram, the height of a bar represents the frequency
of the corresponding data.

We should follow the rules given below while drawing a bar graph.

(i) Each bar should have the same width

(ii) The distance between each pair of bars should be the same.

Now, let’s learn from the given example to draw bar graphs.

Example: There are 30 students in a class. The number of students who
were present on Monday to Friday in the last week are given
below. Draw a bar graph to represent the data.

Days Mon Tue Wed Thu Fri

No. of present students 20 25 28 30 24

A bar graph showing the number of present students of a class in the last week:

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Statistics

EXERCISE 18.2

1. There are 40 students in class VI. The table given below shows the number
of present students in the last week. Draw a bar graph to represent the
data.

Days Sun Mon Tue Wed Thu Fri

No. of present students 30 35 40 37 39 34

2. The table given below shows the number of different animals kept in a zoo.
Draw a bar graph to show their numbers.

Animals Rabbit Monkey Deer Bird Tiger
Numbers 20 16 25 40 5

3. The number of students in the primary level of a school are given below.
Draw a bar graph to show the numbers.

Classes I II III IV V

No. of students 35 30 40 25 20

4. The number of students who passed the S.E.E. examination from a school in
different years are given in the table. Draw a bar graph to represent the data.

Years 2071 2072 2073 2074 2075 2076
25 40 36 48 50 56
No. of students who
passed

5. The bar graph given shows the monthly expenditure of a family in the last 6
months. Answer the following questions.

Questions
a) Write the expenditure of the family in every month.
b) In which months the expenditures were maximum and minimum?

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Statistics

c) If the monthly income of the family is Rs 10,000 every month, express
the expenditures of every month in percent.

d) Express the total expenditure of six months as the percent of the total
income of six months.

6. The bar graph given below shows the S.E.E. results of a school in 5 years.
Answer the following questions.

2072 2073 2074 2075 2076

Questions:
a) In which year the S.E.E. result of the school was the best? What was the

percent of successful students in that year? If 80 students appeared the
exam in that year, how many of them passed?
b) In which year the percent of the successful students was the least? If 50
students appeared the exam, how many of them were unsuccessful?
c) Write a paragraph and discuss about the trend of S.E.E. result of the
school in 5 years.

18.5 Average
Have you ever heard about the following statements?
The average age of the students of class 6 is 11 years.
The average marks of the students in a test in mathematics is 17.
The average temperature of Kathmandu on yesterday was 18qC.
An average is a single number which is used to represent a collection of data.
An average of the given data is calculated by adding them together and dividing
the total by the number of data.

total sum of the data
i.e. average = number of data

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Statistics

Worked-out examples

Example 1: The ages (in years) of 5 children of a class are given below. Find
the average age.

7, 8, 9, 10, 11

Solution: 7 + 8 + 9 + 10 + 11 45
5 5
Here, the average age = = = 9 years

So, the average age of the children is 10 years.

Example 2: The marks obtained by 6 students of class six are 14, 15, 13, 17,

19 and x. If their average marks is 16, find the value of x.

Solution: 14 + 15 + 13 + 17 + 19 + x
6
Here, the average marks = 78 +x
or, 16 = 6

or, 78 + x = 16 u 6

or, 78 + x = 96

or, x = 96 – 78 = 18

Hence, the required value of x is 18.

Example 3: In a class, the average height of 4 children is 90 cm and the
average height of 6 children is 95 cm. Find the average height of
all of these children.

Solution:
Here, the average height of 4 children = Sum of the heights of 4 children
4
Sum of the heights of 4 children
or, 90 = 4

? Sum of the heights of 4 children = 90 u 4 = 360 cm

Again, the average height of 6 children = Sum of the heights of 6 children
6
or, 95 = Sum of the heights of 6 children

? Sum of the heights of 6 children = 95 u 6 = 570 cm. 6

Now, the total number of children = 4 + 6 = 10

And, the total height of 10 children = 360 cm + 570 cm = 930 cm

? The average age of 10 children = 930 = 93 cm.
10
Hence, the required average height is 93 cm.

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Statistics

EXERCISE 18.3
General Section - Classwork
1. Let’s tel land write the answers as quickly as possible.

a) Average of 1 and 3 is ........................
b) Average of 2 and 6 is ........................
c) Average of 1, 2 and 3 is ........................
d) Average of 6, 7 and 8 is ........................

2. a) Bishwant has Rs 10 and Sunayana has Rs 12, they have ........................ in
average.

b) 3 students were absent on Sunday and 5 absent on Monday in two days,
there were ........................ students absent in average.

Creative Section - A

3. a) Define average of the data.

b) Write down the formula of calculating the average of the individual data.

4. Find the average of the following data.

a) 12, 8, 6, 10

b) 20, 40, 25, 15, 30

c) 7 years, 13 years, 10 years

d) 5.5 cm, 8.5 cm, 3.2 cm, 10.8 cm, 7 cm

5. a) The ages of 4 children are 7, 9, 11 and 13 years. Find their average age.

b) The heights of 5 pupils of a class are 80, 85, 90, 95 and 100 cm. Find the
average height of the pupils.

c) The marks obtained by 10 students in mathematics are given below. Find
their average marks.

25, 36, 20, 15, 24, 30, 38, 18, 32, 22

d) The daily wages of 8 workers of a factory are given below. Find their
average wage.

6. a) The marks obtained by 5 students of a class are 18, 24, 20, 14 and x. If
their average marks is 19, find the value of x.

b) The temperature of Kathmandu valley recorded at 6 am. everyday in the
last week was 10qC, 15qC, 4qC, 8qC, 12qC, 10qC and xqC. If the average
temperature was 9qC, find the value of xqC.

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Statistics

c) The average marks of Ram, Hari, Shyam and Gopal in Mathematics is 65.
If Ram, Hari, and Shyam obtained 60, 70 and 80 respectively, how many
marks did Gopal obtain?

d) The average expenditure of a family in the last week was Rs. 111. If the
expenditures on Sunday, Monday, Tuesday, Wednesday, Thursday and
Friday were Rs 140, Rs 126, Rs 105, Rs 70, Rs 84 and Rs 98 respectively,
find the expenditure on Saturday.

Creative Section - B

7. a) The average height of 3 children in a class is 80 cm and the average height
of 5 children is 84 cm. Find the average height of all of these children.

b) In the last year, the average rainfall in Pokhara valley during the first
6 months was 150 mm and the average rainfall of the remaining months
was 60 mm. Find the average rainfall of the whole year.

c) The average selling of the first 4 days of a shop in the last week was
Rs 425 and the average selling of the remaining days was Rs 250. Find
the average selling of the shop in the whole week.

8. a) Shashwat obtained an average marks of 15 in 3 subjects. When the
marks obtained in another subject is also included, the average marks is
increased by 1. Find the marks obtained in the fourth subject.

b) The average marks of English, Nepali, Science and Social Studies
obtained by Sunayana is 70. If the marks obtained in Mathematics is
also included, the average marks is increased by 5. Find her marks in
Mathematics.

It’s your time - Project work

9. a) Let’s make groups of friends and conduct a survey to collect the data

of the number of absentees from classes, 4, 5 and 6 in your school and

find the average number of absentees on (i) Sunday (ii) Sunday and

Monday (iii) Sunday , Monday and Tuesday (iv) in a week.

b) Let’s collect the marks obtained by you and your four friends in a maths
test. Then calculate the average marks.

c) Let’s collect the data of number of family members of your five friends.
Then include the number of your family members and find the average
number of family members.

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