Vedanta Excel in Mathematics Book 6 Final (2078)

Geometry – Angles

2. a) If x° and y° are co-interior angles between parallel lines,
then x° + y° = .................

b) If x° and 80° are co - interior angles between parallel lines,
then x° = ......................

c) If y° and 45° are alternate angles between parallel lines, then y° = ..........
d) If z° and 120° are corresponding angles between parallel lines,

then z° = ....................

Creative Section
3. Find the sizes of unknown angles.

4. Find the sizes of unknown angles.

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Geometry – Angles

5. From the figure given alongside, show that

i) ‘w = ‘c ii) ‘x = ‘s iii) ‘y = ‘g

iv) ‘a = ‘r v) ‘d = ‘s vi) ‘p = ‘c

6. In the adjoining figure, if ‘x + ‘a + ‘y = 180q,
show that ‘a + ‘b + ‘c = 180q.

7. a) In the given figure, ‘ABC = 50q,
‘ACE = 60q and AB//EC. Find the sizes of
‘BAC, ‘DCE and ‘ACB.

b) In the figure alongside, PQ//RS. Find the
values of xq and yq.

c) In the adjoining figure, CE//AB and AC//BF.
Find the sizes of wq, xq, yq and zq.

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Unit Geometry: Triangles and Polygons

14

14.1 Triangle – review

The figure given alongside is a triangle ABC. It is a closed
plane figure bounded by three line segments. They are
called sides of the triangle. AB, BC and CA are the three
sides of triangle ABC. A, B and C are called the vertices of
the triangle. ‘A, ‘B and ‘C are the angles of the triangle.

Triangle ABC can be written as 'ABC. We use a symbol ''' to represent the word
'triangle'.

1. Let’s look at the given triangle ABC, and tell and write the answer in the
blank spaces.

a) The name of triangle is .......................... P

b) Number of sides is ..........................

c) Name of sides are ..........................

d) Number of angles are .......................... Q R
e) Name of angles are ..........................

f) Number of vertices is ..........................

g) Name of vertices are ..........................

14.2 Types of triangles by sides
According to the length of the sides, there are 3 types of triangles.

(i) Equilateral triangle (ii) Isosceles triangle (iii) Scalene triangle

In an equilateral triangle, In an isosceles triangle, the In a scalene triangle,
the lengths of all three lengths of any two sides the lengths of non of the
sides are equal. are equal.
sides are equal.

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Properties of equilateral triangle
(i) In an equilateral triangle, the lengths of its all sides are equal.

? AB = BC = CA.
(ii) The size of each angle of an equilateral triangle is 60q.

? ‘A = ‘B = ‘C = 60q.

Properties of isosceles triangle
(i) In an isosceles triangle, the lengths of its any two sides

are equal.
? AB = AC
(ii) The angles opposite to the two equal sides of an isosceles
triangle are always equal. These equal angles are also
called the 'base angles' of the isosceles triangle.

‘B is the opposite angle of ‘R is the opposite angle ‘X is the opposite angle
side AC and ‘C is opposite of side PQ and ‘P is the of side YZ and ‘Z is the
angle of side AB. opposite angle of side QR. opposite angle of side XY.
Since, AB = AC Since, PQ = QR, Since, XY = YZ

‘B = ‘C. ‘P = ‘R ‘X = ‘Z

14.3 Types of triangles by angles

According to the size of angles, there are 3 types of triangles.

i. Acute angled triangle ii. Obtuse angled triangle iii. Right angled triangle

In an acute angled triangle, its In an obtuse angled triangle, In a right–angled triangle, its
all three angles are acute. its one of the three angles is one of the three angles is a right

obtuse. angle (90q).

Worked-out examples

Example 1: If one of the two acute angles of a right–angled triangle is 65q,
find another acute angle of the triangle.

Solution:

The required acute angle = 90q – given acute angle = 90q – 65q = 25q

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Geometry – Triangles and Polygons

Example 2: In the adjoining figure, find the sizes of the unknown angles.
Solution:

120° x° y°

Here, (i) xq + 120q = 180q [The sum is a straight angle]

or, xq = 180q – 120q = 60q

(ii) xq = yq = 60q [Base angles of isosceles triangle are equal]

? xq = 60q and yq = 60q.

EXERCISE 14.1
General Section – Classwork

1. Let’s tell and write the types of triangles as quickly as possible.

a) All three sides are equal ............................................... triangle.

b) Two sides are equal ............................................... triangle.

c) Non of the sides are equal ............................................... triangle.

d) All three angles are acute ............................................... triangle.

e) One angle is a right angle ............................................... triangle.

f) One angle is an obtuse ............................................... triangle.

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

a) How many acute angles are there in an acute angled triangle ? ............

b) How many obtuse angles are there in an obtuse angled triangle? ............

c) Are two obtuse angles possible in a triangle ? ............

d) How many right angles and acute angles are there in a right angled tri-

angle? ............

e) Is an obtuse angle possible in a right angled triangle ? ............

f) If one of the acute angles of a right angled triangle is 50°, the size of

remaining acute angle is ............

Creative Section

1. Find the sizes of unknown angles of these triangles.

a) b) c) d)

x° x° 60°

40° 55° y°

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Geometry – Triangles and Polygons

e) f) g) h)

x 2x y°

(x+20)° 30° 140° a° b°

x° 45°

2. Find the sizes of unknown angles of these triangles.

72°
a°
50° 115°

40°

108°
p° m°

q° n° 48°

14.4 Sum of the angles of a triangle

Let’s measure the size of each angle of the following triangle. Copy and complete
the table given below:

(i) ‘BAC (ii) (iii)
70q
Fig ‘ABC ‘ACB ‘ABC + ‘BAC + ‘ACB
(i) 50q ……… 60q 50q + 70q + 60q = 180q
(ii) ……… ………
(iii) ……… ……… ………………………
……… ………………………

Thus, sum of the three angles of a triangle is always 180q.

14.5 Exterior angle of a triangle

In the adjoining triangle, aq, bq and cq are called the
interior angles of 'ABC. Let's produce the side BC to
D. Now, xq is the angle made by the produced side and
one side of the triangle.

xq is called the exterior angle of the triangle.

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Geometry – Triangles and Polygons

Again,

(i) aq + bq + cq = 180q [Sum of the angles of a triangle]

(ii) xq + cq = 180q [The sum is a straight angle]

? xq + cq = aq + bq + cq [from (i) and (ii)]

or, xq = aq + bq + cq – cq

or, xq = aq + bq

Thus, the exterior angle of a triangle is equal to the sum of two opposite interior
angles which are not adjacent to it.

Worked-out examples

Example 1: If yq, 2yq and 3yq are the angles of a triangle find them.

Solution:

Here,yq + 2yq + 3yq = 180q [Sum of the angles of a triangle is 180q]

or, 6yq = 180q
180°
or, yq = 6 = 30q

? y = 30q, 2yq = 2 u 30q = 60q and 3yq = 3 u 30q = 90q.

Example 2: In the adjoining figure, find the size of

unknown angles. 110°
Solution:

Here,xq + 110q = 180q [The sum is a straight angle.]

or, xq = 180q – 110q = 70q

Also, yq = xq = 70° [Base angles of an isosceles triangle]

Again,

xq + yq + zq = 180q [The sum of the angles of a triangle]

or, 70q + 70q + zq = 180q Alternative process
or, 140q + zq = 180q
or, zq = 180q – 140q y+z = 110
or, zq = 40q
? xq = 70q, yq = 70q and zq = 40q x + y + z = 180°

x = 180° – 110° = 70°

y = x = 70°

z = 110° – 70° = 40°

EXERCISE 14.2
General Section – Classwork
1. Let’s tell and write the answers as quickly as possible.

a) If x°, y° and z° are the angles of a triangle, then x° + y° + z° = .................
b) If x°, 110° and 40° are the angles of a triangle, then x° = .....................
c) If x° is the exterior angle and y° and z° are the two opposite interior angles

of a triangle, then x° = .............................

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Geometry – Triangles and Polygons

d) If p° is the exterior angle and 40° and 60° are the two opposite interior
angles of a triangle, then the size of p° = ....................

e) x°, y° and z° are the tree angles of a triangle. If x° = y° = 50°, the sized of
z° = ....................

f) a°, b° and c° are the three angles of a triangle. If a° + b° = 140°, the size of
c° = ..........................

Creative Section
2. a) a°, 2a° and 60° are the angles of a triangle. Find the size of a° and 2a°.

b) If p°, 2p° and 3p° are the angles of a triangle, find them.
c) If the angles of a triangle are in the ratio 2:3:4, find them.
d) If the angles of a triangle are in the ratio 5:6:7, find them.
3. a) If x° is the exterior angle and 30° and 50° are the two opposite interior

angles, find the value of x°.
b) If 110° is the exterior angle and 70° and y° are the opposite interior angles,

find the value of y°.
c) If 2x° is the exterior angle and x° and 45° are the opposite interior angles,

find the value of x°.
4. Find the sizes of unknown angles of these triangles.

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Geometry – Triangles and Polygons

m) n) o) p)

55° m°
75°a°
62° a°
c° b° x° 78°

y°

It’s your time: Project work
5. a) Measure the sizes of angles of each of these triangles and complete the

table. Also write your conclusion.

∠A ∠B ∠C ∠A + ∠B + ∠C ∠P ∠Q ∠R ∠P + ∠Q + ∠R

Conclusion .....................................................................................................

b) Draw two triangles of different shape and size. Measure each of the angles
and add them, what conclusion do you draw out?

14.6 Quadrilaterals

A quadrilateral is a closed plane figure bounded by four
line segments. In the figure, ABCD is a quadrilateral. Here,
AB, BC, CD, and DA are the sides of the quadrilateral. A,
B, C and D are the vertices of the quadrilateral. ‘A, ‘B,
‘C and ‘D are the angles of the quadrilateral.

14.7 Some special types of quadrilaterals

Rectangle, square, parallelogram, rhombus, trapezium, etc. are some special
types of quadrilaterals.

(i) Rectangle

In the figure, ABCD is a rectangle. Its opposite sides
are equal and each of the four angles is 90q.
So, AB = DC, BC = AD, AB//DC and BC//AD

‘A = ‘B = ‘C = ‘D = 90q.

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Geometry – Triangles and Polygons

(ii) Square

In the figure, PQRS is a square. Its all sides are equal and
each angle is 90q.
So, PQ = QR = RS = SP and ‘P = ‘Q = ‘R = ‘S = 90q
(iii) Parallelogram

In the figure, ABCD is a parallelogram. Its
opposite sides are equal and parallel. Its
opposite angles are equal.

So, AB = DC, BC = AD, AB//DC and BC//AD
‘A = ‘C and ‘B = ‘D.

(iv) Rhombus

In the figure, WXYZ is a rhombus. It is a parallelogram
having all sides equal. Its opposite angles are equal.

So, WX = XY = YZ = ZW and WX//ZY and XY//WZ
‘W = ‘Y and ‘X = ‘Z.

(v) Trapezium

In the figure, PQRS is a trapezium. Its any two
opposite sides are parallel. So, PQ//SR.

14.8 Tangram
A tangram consists of seven polygons called tans, which are arranged together
to form different plane shapes without overlapping.
We can make a tangram by folding a square sheet of paper as shown in the
diagrams given below.

Now, let’s make a rectangle, square, parallelogram, trapezium and rhombus by
using the 7 polygonal pieces.

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Geometry – Triangles and Polygons

14.9 Sum of the angles of a quadrilateral
Let’s measure the size of each angle of the following quadrilaterals and complete
the table given below.

Fig. ‘A ‘B ‘C ‘D ‘A + ‘B + ‘C + ‘D
(i) 90q 90q 90q 90q 90q + 90q + 90q + 90q = 360°

(ii) …….. …….. …….. …….. ……..……..……..……..

(iii) …….. …….. …….. …….. ……..……..……..……..

Thus, the sum of the four angles of a quadrilateral is always 360q.

14.10 Polygons
Let’s observe the shape of the following plane figures.

A polygon is a closed plane figure bounded by three or more line segments.
Triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, etc. are a
few examples of polygons.

The table given below shows the different types of polygons, their number of
sides and names.

Polygon Number Name of Polygon Number Name of
of sides polygon of sides polygon

3 Triangle 6 Hexagon

4 Quadrilateral 7 Heptagon
5 Pentagon 8 Octagon

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Regular polygon
Polygons having equal length of sides and the same size of interior angles
are called regular polygons. Equilateral triangle and square are also regular
polygons.

We can calculate the sum of the angles of a polygon by using a formula.
Sum of the angles of a polygon = (n – 2) u 180q,
where, ‘n’ is the number of sides of polygons.

Worked-out examples

Example 1: If xq, 2xq, 3xq and 4xq are the angles of a quadrilateral, find them.

Solution: I have remembered!
Sum of the angles of a
Here,xq + 2xq + 3xq + 4xq = 360q quadrilateral is 360°.

or, 10xq = 360q
360°
or, xq = 6 = 36q

Now, 2xq = 2 u 36q = 72q, 3xq = 3 u 36q = 108q and 4xq = 4 u 36q = 144q.

So, the required angles of the quadrilateral are 36q, 72q, 108q and 144q.

Example 2: In the adjoining figure, find the angles of
the parallelogram.

Solution:

Here, wq = 75q [Being corresponding angles]

yq = wq = 75q [Being opposite angles of the parallelogram.]

zq + 75q = 180q [The sum is a straight angle]
or, zq = 180q – 75q = 105q

xq = zq = 105° [Being opposite angles of the parallelogram]

So, the required angles of the parallelogram are, wq = yq = 75q and xq = zq = 105q.

EXERCISE 14.3
Let’s tell and write the correct answers as quickly as possible.
1. a) The size of each angle of a rectangle is ............................

b) The size of each angle of a square is ..............................

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Geometry – Triangles and Polygons

c) If a side of a rectangle is 4.5 cm, the length of its opposite side is ............

d) If x° and 65° are the two opposite angles of a parallelogram, then the
size of x° is ...............

e) If a°, b°, c° and d° are the angles of a quadrilateral, then the size of

a° + b° + c° + d° = ......................

f) If w°, x°, y° and z° are the angles of a quadrilateral and

w° + x° + y° = 300°, then the size of z° is .......................

2. a) The number of sides of a pentagon is ...........................

b) The number of sodes of an octagon is ...........................

c) A six sided polygon is called a ...........................

d) A seven sided polygon is called a ...........................

e) The sum of the angles of polygon is

obtained by using the formula ...........................

Creative Section - A AD

3. a) Write down the name of given polygon. Also, write

down the name of 2 triangles which is formed inside it. B SC
b) Write down the name of the given polygon. Also, write

down the name of 3 triangles and 2 quadrilaterals which T R

are formed inside it.

c) From the figure given alongside, write down PQ
(i) name of polygon SE

(ii) name of 4 triangles HM

(iii) name of 3 quadrilaterals RA
(iv) name of 2 pentagons

4. a) If x°, 75°, 105° and 80° are the angles of a quadrilateral, find the size
of x°. Hint : x° + 75° + 105° + 80° = 360°

b) If p°, 2p°, 90° and 120° are the angles of a quadrilateral, find the size
of p° and 2p°.

c) If x°, 2x°, 3x° and 150° are the angles of a quadrilateral, find the size
of x°, 2x° and 3x°.

d) If 2a°, 3a°, 4a° and 6a° are the angles of a quadrilateral, find them.

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Geometry – Triangles and Polygons

5. Find the unknown sizes of angles of these quadrilaterals.

z°

Creative Section - B

6. Find the sum of the interior angles of the following polygons.
a) pentagon b) hexagon c) heptagon d) octagon

7. a) If x°, 80°, 120°, 105° and 135° are the angles of a pentagon, find x°.
b) If p°, 2p°, 3p°, 4p° and 5p° are the angles of a pentagon, find them.
c) If y°, 4y°, 100°, 150°, 120° and 160° are the angles of a hexagon, find the
sizes of unknown angles.
d) If x°, 2x°, 3x°, 4x°, 140° and 160° are the angles of a hexagon, find the
sizes of unknown angles.

8. Find the unknown sizes of an angles of the following polygons.

It’s your time: Project work

9. a) Draw a pentagon in your own choice. Measure each of its angles and
verify the result using formula.

b) Draw a pentagon in your own choice. Measure each angles, find the
sum and verify the result using formula.

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Geometry – Triangles and Polygons

14.11 Circle
1. Let’s look at the following diagrams and answer the questions as quickly

as possible.

Nanglo Coin Watch Button

a) The shape of face of ‘Nanglo’ is ………………...

b) The shape of face of coin is ………………...

c) The shape of face of watch is ………………...

d) The shape of button is ………………...

2. Let’s discuss and list 5 more objects available in your surroundings which
have the circular faces.

3. Let’s take a compass with a pencil and fix the compass needle at
a point in a sheet of paper. Then, take an arc of suitable radius
in the compass and rotate it about the fixed point through a
complete turn. Observe the shape so formed and name it.

4. Let’s draw a closed curved around a coin or bangle or a bowl on a sheet of
paper and discuss about the shapes that you have drawn.

Circle is a plane figure bounded by a curved line and its every point is
equidistant from a fixed point. The fixed point is called the centre of the
circle.

(i) Parts of circle B

Circumference of a circle OA
The curved boundary line of a circle is called its Radius (r)
circumference. The length of the circumference represents
the perimeter of the circle.

Radius of a circle
The straight line segment that joins the centre of a circle to a
point on its circumference is called the radius of the circle.
In the figure given alongside, OA is a radius of the circle.
Radii (plural of radius is radii) of the same circle are always
equal. OA = OB = r

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Chord of a circle O Chord
The straight line segment that joins any two points on the A B
circumference of a circle is called the chord of a circle.
In the given figure, AB is the chord of the circle.

Diameter of a circle

The chord that passes through the centre of a circle is called
the diameter of the circle. Diameter is the longest chord of
any circle.
C O D
Diameter
In the figure, CD is the diameter of the circle. The length of
the diameter of the circle is two times its radius.

? Diameter = 2 u radius.

Sector of a circle O
The region inside a circle bounded by its two radii is called AB
the sector.
In the adjoining figure, the shaded region AOB is the sector.

Arc of a circle A
The part of a curve between two given points on the curve of Arc O
a circle is called an arc.
In the figure alongside, AB is the arc. B

Segment of a circle O B
The region bounded by an arc and its corresponding chord A
is called the segment of a circle.
In the given figure, the shaded region represents a segment.

Semi-circle B
AC
Half part of a circle is called a semi-circle. A diameter divides
a circle into two halves and each half is the semi-circle. O

In the given figure, ABC is a semi-circle.

(ii) Some important facts about circle A C
a) The radii of circle are always equal. O

A B

OC Figure (ii)
B

Figure (i)

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Geometry – Triangles and Polygons

Let’s measure the radii OA, OB and OC in each of the above figures and write
the results in the table below.

Figure OA Radii OC Result
(i) ...................... OB ...................... ......................

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

(ii) ...................... ...................... ...................... ......................

Conclusion: ..........................................................................................................

b) The diameters of circle are always equal. E D
A
F CO
D

A OB

C B
E F

Figure (i) Figure (ii)

Let’s measure the diameters AB, CD and EF in each of the above figures and

write the results in the table below.

Figure AB Diameters EF Result
CD

(i) ...................... ...................... ...................... ......................

(ii) ...................... ...................... ...................... ......................

Conclusion: ..........................................................................................................

c) The diameter of a circle is always two times its radius. B

D

A OB CO

CA
D

Figure (i) Figure (ii)

Let’s measure the diameter AB and radii OC and OD in each of the above figures

and write the results in the table below.

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Geometry – Triangles and Polygons

Figure Diameter Radii Result
AB OC OD

(i) ...................... ...................... ...................... ......................

(ii) ...................... ...................... ...................... ......................

Conclusion: ..........................................................................................................

EXERCISE 14.4

1. Let’s look at the given circle, say and write the correct names of its parts.

a) O is the ……………… C
b) OC is a ……………… Q
c) OA, OB and OC are ………………
AOB

DE
P

d) AB is a ………………

e) DE is a ………………

f) AB and DE are ………………

g) Region BOC bounded by the radii OB and OC with the arc BQC
is a ………………

h) Region DPE bounded by the chord DE with the arc DPE is a ………………

2. Let’s write ‘True’ or ‘False’ for the following statements.

a) Circle is a plane figure. .................

b) A circle has exactly one centre. .................

c) Radius is the line segment that joins any two .................
points of the circumference.

d) All the radii of a circle are equal. .................

e) Every chord is a diameter. .................

f) Diameter divides the circle into two halves. .................

g) The diameter of the circle is double of its radius. .................

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h) A diameter is the longest chord in a circle. .................

i) Sector is the part of circle enclosed within two .................
radii and their corresponding arc.

j) The segment of a circle is the region between .................
the chord and corresponding arc.

Creative Section

3. a) What do you mean by circle? Give any five real life examples of circular
objects.

b) Define circumference of a circle. Also, show it in a diagram.
c) Define radius and diameter of a circle with a diagram.
d) What is the relation between the diameter and radius of a circle? Find

the length of a diameter of circle whose radius is 5 cm.
e) In what condition, a chord of a circle becomes a diameter?
f) Differentiate between sector and segment of a circle with an appropriate

diagram.
4. a) How many radii can you draw in a circle?

b) How many chords can be drawn in a circle?
c) Draw as many circles with the same centre.

d) How many circles can be drawn through two given points?

It’s time- Project work

5. a) Let’s take a square sheet of paper. Fold it to get the following shapes.
Cut out the folded paper of the fifth step along the dots shown in (v)
and unfold it. Write a short note about the shape that you have got.

(i) (ii) (iii) (iv) (v)

b) Let’s draw a circle on a chart paper and cut it out. Fold the circular
paper into half and again fold into half. Unfold the paper and find the
number of sectors you made. Also, colour one of the sectors.

c) Let’s draw as bigger circle as possible on a chart paper and show its
centre, circumference, radius, chord, diameter, sector and segment with
different colours and present in classroom.

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Unit Geometry: Construction

15

15.1 Construction of angles
We can use protractor or compasses to construct angles.

1. Construction of angles by using protractor

We use a protractor to measure as well as to construct the given angles. Here,
let’s learn to construct a given angle by using protractor.

Construct ‘AOB = 70q. B
A
Steps

(i) Draw an arm OA, and place your
protractor as shown.

(ii) Count round the edge from 0q to
50q, and mark B.

(iii)Remove the protractor, and join
OB.

Now, you have constructed ‘AOB = 70q

2. Construction of 30°, 45°, 60° and 90° using set-squares.

Let’s take out the set-squares from your own geometry box and learn to
construct the following angles.

8

45° 60° 7

6
5

4

3

90° 45° 2 B
8 9 10 11 1
30° 90°1 2 3
45 67

Construct ‘AOB = 30q. 60°

Steps O 30° 90° A
(i) Draw a line segment

(ii) Place the 30° corner of a (30°, 60°, 90°) set-square at the point of the line
segment OA.

(iii)Draw line segment OB. ‘AOB = 30° is the required angle.

Similarly, draw an angle of 45°, 60° and 90° by using set-squares.

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Geometry – Construction

2. Construction of angles by using compasses
We can also use compasses to construct certain standard angles, such as:
30q, 60q, 90q, 120q, etc.

a) Construct ‘AOB = 60q.
Steps
(i) Draw an arm OA and place the pointed metal end of the compasses

at O.
(ii) Draw an arc taking suitable radius to cut OA at C.
(iii)Place the pointed metal end of the compasses at C and draw an arc of the

same radius to cut the first arc at D.
(iv) Join O, D and produce it to B.

Now, ‘AOB = 60q is the required angle.

b) Construct ‘AOF = 120q

Steps:

(i) Repeat the process of construction of 60q till step 2 a) (iii)
(ii) Now, place the metal end of the compasses at D and draw an arc of the

same radius to cut the first arc at E.
(iii) Join O, E and produce it to F.

Here, ‘AOF = 120q is the required angle.

c) Construct ‘AOG = 90q.

Steps

(i) Repeat the process of construction of 120q till step 2 b) (ii).
(ii) Now, place the metal end of the compasses at D and E respectively and

draw two arcs of the same radius cutting each other at G.
(iii) Join O, G.

Here, ‘AOG = 90q is the required angle.

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G

Similarly,

(i) Place the metal end of the compasses on the first arc
at 60q and 90q respectively and construct 75q.

(ii) Place the metal end of the compasses on the first
arc at 90q and 120q respectively and construct
105q.

(iii)Place the metal end of the compasses on the
first arc at 120q and 180q respectively and
construct 150q.

15.2 Bisecting a given angle

When an angle is divided into two halves, it is said to be bisected. Now let’s
learn the process of bisecting a given angle.
Construct an angle of 60q and bisect it.

Steps:

(i) Construct the given ‘AOB = 60q.

(ii) Place the metal end of the compasses at the point
C and D respectively and draw two arcs of suitable
radius. Let the arcs intersect each other at E.

(iii)Join O, E. 1 1
2 2
Here, ‘AOE = ‘BOE = 30° = of 60q or ‘AOB.

EXERCISE 17.1

1. Let’s construct the following angles by using protractor.

a) 20q b) 40q c) 50q d) 65q e) 70q f) 85q
l) 165q
g) 100q h) 110q i) 130q j) 145q k) 150q

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2. Let’s construct the following angles by using compasses.

a) 60q b) 120q c) 90q d) 75q e) 105q

f) 150q g) 135q h) 45q i) 30q j) 165q

3. Let’s construct the following angles and bisect them.

a) 60q b) 120q c) 90q d) 30q e) 45q

f) 75q g) 150q h) 105q i) 15q j) 165q

4. Let’s draw the following angles by using protractor and bisect them.

a) 20q b) 40q c) 50q d) 70q e) 80q

f) 100q g) 110q h) 130q i) 140q j) 160q

15.3 Construction of perpendiculars
A straight line segment is said to be perpendicular to a given straight line segment
if it makes an angles 90q to the given line segment. We can draw perpendiculars
either by using set–squares or by using compasses.

1. Construction of perpendicular by using set–squares

a) At first, let’s draw a perpendicular to a line from a
point outside the line.
Steps

(i) Draw a line segment AB and mark a point P just
above it.

(ii)Place the two set squares as shown in the figure
and draw a line from the point P on AB.

Here, PQ is perpendicular to AB at Q.

b) Now, let’s draw a perpendicular to a line at a point on the line.
Steps
(i) Draw a line segment AB and mark a point P on it.

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(ii) Place the two set squares like before as shown in the figure and draw
a line on the point P.
Here, QP is perpendicular to AB at P.

2. Construction of perpendiculars by using compasses

a) At first, let’s draw a perpendicular to a line from a point outside the
line.

Steps

(i) Draw a line segment AB and mark a point P
just above it.

(ii) Place the metal end of the compasses at P
and draw an arc to cut AB at X and y.

(iii)Now, place the metal end of the compasses
at X and Y respectively and draw two
intersecting arcs just below the line. Let
the point of intersection of these arcs be O.

(iv) Join P, O that meets AB at Q.
Here, PQ is perpendicular to AB at Q.

b) Now, let’s draw a perpendicular to a line at a point on the line.
Steps
(i) Draw a line segment AB and mark a point P
on it.
(ii) Place the metal end of the compasses at the
point P and draw an arc to cut AB at X and
Y.
(iii)Place the metal end of the compasses at X
and Y respectively and draw two arcs of the
same radius to cut the first arc at M and N.

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(iv) Now, place the metal end of the compasses at M and N respectively
and draw two intersecting arcs. Let the point of intersection of these
two arcs be Q.

(v) Join, P, Q.

Here, PQ is perpendicular to AB at P.

3. Construction of perpendicular bisector of a given line

A straight line segment which is perpendicular to a given line segment and
divides it into two equal halves is called the perpendicular bisector of the
line.

Let’s learn the process of drawing the perpendicular bisector of a line.

Step

(i) Draw a straight line segment AB of any length.

(ii) Place the metal end of the compasses at the point
A and B respectively and draw two intersecting
arcs above and below the line. Let P and Q are
the points of intersecting of these arcs.

(iii)Join P, Q.

Here, PQ is the perpendicular bisector of the line
AB.

15.4 Construction of a line parallel to given line

We can construct a line parallel to a given line
either by using set squares or by using compasses.
1. Construction of a line parallel to a given line

by using set squares

Steps

(i) Draw a line segment AB and mark a point
P just above it.

(ii) Place an edge of a set square along the
line.

(iii)Place an edge of another set square along
the edge of the first set square and the
point as shown in the figure.

(iv) Move the first set square slowly up to
the point P holding the second set square
firmly.

(v) Now, draw PQ as shown in the figure.

Here PQ is parallel to AB.

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2. Construction of a line parallel to a given line by using compasses
Steps
(i) Draw a line segment AB and mark a point P
just above it.

(ii) Join P, A.

(iii)Place the metal end of the compasses at A and
draw an arc cutting AB at C and AP at D.

(iv) Place the metal end of the compasses at P and draw the same arc to cut
AP at E.

v) Measure the length of the arc CD with the help of the compasses and cut
the same length of arc EF from E.

(vi) Join PF and extend to both directions.
Here, QR is parallel to AB.

EXERCISE 15.2

1. Let’s draw the following line segments. Mark any point just above each line
segment and draw perpendicular from the point to the line by using set
squares.

a) AB = 6 cm b) PQ = 5.5 cm c) XY = 3.8 cm d) MN = 6.7 cm

2. Let’s draw the following line segments. Mark any point on each line segment
and draw perpendicular on the point to the line by using compasses.

a) AB = 5 cm b) PQ = 4.5 cm c) XY = 6.4 cm d) MN = 3.8 cm

3. a) In the given figure, A

(i) Is ED a perpendicular bisector of BC? FE
(ii) Is EF a perpendicular bisector of AB?
b) In the figure given alongside, BD C
(i) What is the perpendicular bisector of XZ?
WZ

(ii) What is the perpendicular bisector of WY? XY

4. Let’s draw the perpendicular bisector of each of the following line segments.

a) AB = 4 cm b) CD = 5 cm c) PQ = 6.8 cm d) RS = 5.2 cm

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5. Let’s draw the following line segments. Mark any point just above each line
segment and draw a line parallel to the given line by using set squares.

a) AB = 6.3 cm b) CD = 4.5 cm c) P = 5.6 cm d) RS = 3.7 cm

6. Draw the following line segments. Mark any point just above each line
segment and draw a line parallel to the given line by using compasses.

a) AB = 6.4 cm b) XY = 4.8 cm c) PQ = 3.6 cm d) MN = 5.9 cm

15.5 Construction of regular polygons
A polygon is said to be the regular polygon if the length of its all sides are equal
and the size of each angle is also equal. Equilateral triangle, square, regular
pentagon, regular hexagon, etc. are a few examples of regular polygons.

1. Construction of equilateral triangle
In an equilateral triangle, its all sides are equal and each angle is 60q.
Let’s learn to construct an equilateral triangle.
Construct an equilateral triangle
whose each side is 4.5 cm.

Steps
(i) Draw AB = 4.5 cm
(ii) Construct ‘A = 60q and ‘B = 60q at

A and B respectively.
(iii)Let C be the point of intersection of

the arms of ‘A and ‘B.
Here, 'ABC is the required equilateral
triangle.

2. Construction of square
All sides of a square are equal and its each
angle is 90q.
Construct a square whose each side is 3.4 cm.
Steps
(i) Draw PQ = 3.4 cm
(ii) Construct ‘P = 90q and ‘Q = 90q at P
and Q respectively.
(iii)On PX and QY, mark PS = QR = 3.4 cm
(iv) Join S, R.
Here, PQRS is the required square.

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3. Construction of rectangle

Opposite sides of a rectangle are equal
and its each angle is 90°.

Construct a rectangle whose length is
5.5 cm and breadth is 4 cm.
Steps
(i) Draw AB = 5.5 cm
(ii) Construct ∠A = 90° and ∠B = 90°

at A and B respectively.
(iii) On AX and BY, mark AD = BC = 4 cm.
(iv) Join C and D
Here, ABCD is the required rectangle.

EXERCISE 15.3

1. The length of the sides of different equilateral triangles are given below.
Construct them.

a) 'ABC, AB = 5 cm b) 'PQR, PQ = 4.5 cm c) 'XYZ, XY = 3 cm

d) 'KLM, KL = 4.5 cm e) 'DEF, DE = 6 cm f) ' RST, RS = 5.5 cm

2. The length of the sides of different squares are given below. Construct them.
a) ABCD, AB = 4 cm b) PQRS, PQ = 3 cm c) WXYZ, WX = 4.6 cm
d) EFGH, EF = 3.5 cm e) KLMN, KL = 5 cm f) QRST, QR = 5.3 cm

3. Construct rectangles whose lengths and breadths sre given below.
a) length = 5 cm, breadth = 4 cm b) length = 6.4 cm, breadth = 3.8 cm
c) length = 4.7 cm, breadth = 3.5 cm d) length = 3 cm, breadth = 5 cm.

It’s your time - Project work!

4. Let’s draw two equilateral triangle of your own measurements. Draw the
perpendicular bisector of a side of each triangle. Does it pass through the
opposite vertex in each equilateral triangle? Discuss your investigation with
your friends.

5. Let’s draw two squares of your own measurements. Draw two diagonals in
each square. Do these diagonals bisect each other perpendicularly? Discuss
your findings with friends.

6. Draw two rectangles of your own measurements. Draw two diagonals in each
rectangle. Do these diagonals bisect each other in each rectangle? Discuss
your findings with friends.

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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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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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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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? 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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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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Perimeter, Area and Volume

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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Perimeter, Area and Volume g) h)

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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