Prime Math 6 - Final rabi_final

Unit Revision Test II

1. Define: (a) a rectangle (b) a regular hexagon

2. Find the sum of the angles of a pentagon.

3. Find the number of sides of the polygon having sum of interior angles 540o.

4. Find the number of sides of the regular polygon having size of an interior
angle 135o.

5. From the figure given alongside, find the size of unknown angle x.

132o 115o
120o

x 100o

6. Construct:
(a) An equilateral triangle of side 6cm
(b) A square of side 5cm

Prime Mathematics Book - 6 45

Estimated periods – 6

Objectives

At the end of this unit, the students will be able to:

know about the ordered pairs.

give the introduction of axes, quadrants and co-ordinates.
find the co-ordinates of the given point in the graph.

plot the given points in the graph. Y

+5
+4

+3

+2

X' -5 -4 -3 -2 -1 +1 X

-1 o+1 +2 +3 +4

-2

-3
-4

-5

Y'

Teaching Materials Activities

Graph chart, geoboard, different colour It is better to:
sign pen, scale, pencil. demonstrate the graph with axes and
origin in the graph paper.
display the graph chart with axes and
origin to give the concept of plotting the
points on the graph paper.
to give the idea to find the co-ordinates
of the point on the graph paper.

Ordered pairs

6

5 House Bus Park
Tree Hospital
4
3 School Pond

2 Temple

1 Petrol
pump

0 12345678
To find the position of the house in the above figure, first we count 3 units towards
right along the horizontal from 0 and then count 5 units up wards from 3. House lies
at (3, 5). Similarly, we can say the position of temple is (1, 2), petrol pump is (3,1)
etc

So, we use a pair of numbers within brackets by using comma (,) to fix the position
of place.

When we change the order of the numbers, the positions of the places are also
changed. For example, in the above figure, (3, 1) is the position of petrol pump.
When 3 and 1 are exchanged, the position is (1, 3) which is the position of school.
So, the orders of these number pairs are fixed which are called ordered pairs. We
use this ordered pairs to indicate the position of a point in a plane.

2.1 Co- ordinates: Y

In the figure given alongside, XOX' and YOY' are +5
two number lines which are perpendicular to +4
each other at the point O. These number lines
are called the co-ordinate axes. The number line +3
XOX' (horizontal line) is called x- axis and the
number line YOY' (vertical line) is called y-axis. +2

X' -5 -4 -3 -2 -1 +1 X

-1 o+1 +2 +3 +4

-2

-3
-4

-5

Y'

Prime Mathematics Book - 6 47

The point of intersection of XOX' and YOY' i.e. O is called the origin.

In the above figure, XOX' and YOY' divide the plane figure into four parts and each
part is in the form of open rectangles. So each part is called the quadrants. The four
quadrants are XOY, X'OY, X'OY' and XOY'.

1st quadrant XOY OX- positive, OY – positive (+, +)

2nd quadrant X'OY OX'- negative, OY – positive (-, +)

3rd quadrant X'OY' OX'- negative, OY' – negative (-, -)

4th quadrant XOY' OX - positive OY' – negative (+, -)

To find the position of a point on the plane, we Y
draw the perpendiculars on the x-axis and Y- axis
from the point. So, in the adjoining figure, for the NA
position of a point A, we draw the perpendiculars
AM on the X-axis and AN on the Y-axis from the X' o X
point A. The distance OM = AN along X-axis is called M
the X-coordinate and the distance AM = ON along
Y-axis is called the Y- coordinate of the point A. Y'
In the figure, OM is 3 units and ON is 3 units. Thus
the X-coordinate of the point A is 3 and Y-coordinate
of the point A is 3. Therefore, the co-ordinates of
the point A is (3, 3).

The position of the point A is (3, 3). But (3, 3) is an ordered pair. So, the ordered
pairs give the position of a point on the plane.

Thus, X-coordinate of a point means the perpendicular distance of the given point
from the Y-axis. Y-coordinate of a point means the perpendicular distance of the
point from the X-axis.

The co-ordinates system which gives the accurate position of the point in the plane
was first invented by a French mathematician Rene Descrates in the 17th century.

48 Prime Mathematics Book - 6

Plotting the points: X' Y

We can plot a point in the cartesian plane with oX
the help of its co-ordinates . A(3, -4)

On the graph, draw the co-ordinate axes XOX' Y'
and YOY'. Then take a suitable scale and mark
on the axes. Positive values are marked along
OX and OY. Similarly, negative values are marked
along OX' and OY'.

For plotting a point A whose position is (3,-4). First we take the number 3 of the
ordered pair (3,-4) along the X-axis towards the positive direction. Then we take
the number -4 of the ordered pair (3,-4) along the Y-axis towards the negative
direction which is shown in the adjoining figure.

Note: - The co-ordinate of the origin O is (o, o)
- The co-ordinates of a point on the x-axis is (x, o),where x is the value of
x-coordinate of the point counting form the origin along x-axis.
- The co-ordinates of a point on the y-axis is (o, y), where y is the value of
y-coordinate of the point counting form the origin along y-axis.

Exercise 2.1

1. Fill in the blanks:

a) If the co-ordinates of a point are (4, 5), the x-coordinate and Y-coordinate
are…………. and …………. respectively.

b) If the co-ordinates of a point are (-3, 0), the point lies on ……………..

c) If x co-ordinate of a point is 3 and Y-coordinate is 2, the position of the
point is …………………..

d) If the co-ordinates of a point is (3, 4), the point lies on ……….quadrant.

e) A point is 5 units right along x-axis and 6 units down along Y-axis from
the origin. The position of the point is ……………..

Prime Mathematics Book - 6 49

2. Find the co-ordinates of the points A, B, P, Q, M, N and R from the graph
given alongside.

Y
Q

A

NP

X' o X

M
B

R Y'

3. Plot the following points in the graph.

A(2, 3), B(-4, 3), C(4, -2), D(0, 4), E(-4, -5)
I (0, -3), J (-7, 6)
F(-3, 0), G(6, -5), H (5, 3),

4. Plot the following points and join them in order by using ruler. Write the
name of the figures so formed.

(a) A (2, 3), B(4, 5), C (6, 1)

(b) E(0, 2), F(4, 1), G(3, -6)

(c) A(0, 4), B(4, 0), C(0, -4), D(-4, 0)

(d) P(0, -2), Q(-4, 2), R(-6, 0), S(-4, -5)

5. (a) A(2, 3), B(2, -2) and C(7, -2) are the vertices of a rectangle. If D is the
fourth vertex of the rectangle, find the co-ordinates of vertex D by
plotting the given vertices in the graph.

(b) Plot the points A(2, 4) and B(5, 1). Join A and B and produce it both ways
cutting the X and Y axes at the points M and N respectively. Find the
co-ordinates of the points M and N

(c) Plot the points P (4, 3) and Q (12, -5). Join P and Q and find the
co-ordinates of the mid-point of PQ.

50 Prime Mathematics Book - 6

6. Write down the co-ordinates of the vertices of the given different geometrical
shapes in the following graph.

(a) S Y B
X' C
P Q R
M P A X

NO

Q Y'

Unit Revision Test I
1. Fill in the blanks:

a) If the co-ordinates of a point are (2, -3), the x-coordinate and Y-coordinate
are…………. and …………. respectively.

b) If the co-ordinates of a point are (2, 0), the point lies on ……………..

c) If the co-ordinates of a point is (3, -4), the point lies on ……….quadrant.

2. Plot the following given points in the graph.

A(2, -3), B(4, 5), C(-3, 2), D(5, 0), E(-3, -7)

3. Plot the following points and join them in order by using ruler. Write the
name of the figures so formed.

(a) A (3, 2), B(5, 4), C (7, -1) D(-3, 0)
(b) A(0, 3), B(3, 0), C(0, -3), S(-3, -5)
(c) P(0, -3), Q(-3, 2), R(-5, 0),

4. (a) Plot the points A (5, 2) and B (9, -4). Join AB and find the co-ordinates
of the mid point of AB.

(b) A (-4, 1), B (-2,1), C (0, 3) are three vertices of a parallelogram ABCD. If
the fourth vertex D is opposite of vertex B, find the co-ordinates of the
vertex D by plotting the given vertices in the graph.

Prime Mathematics Book - 6 51

Estimated periods – 12

Objectives

At the end of this unit, the students will be able to:
give the introduction about perimeter, area and volume of the closed figures.
find the perimeter of the triangle, rectangle and regular closed figures.
find the area of the regular and irregular closed figures by counting square
boxes.
find the area of the rectangle and square by using formula.
find the voulme of cuboids and cubes.

1.5cm 1.5cm 2cm
3cm
3cm 1cm 2cm

4cm 12cm

6cm 4cm
6cm
15cm

Teaching Materials Activities

Models of different solid objects like It is better to:
triangle, rectangle, square, cube and demonstrate the edges, corners and
cuboid, scale, pencil, different colour sign surface of the different solid objects.
pen, chart of the formula. display the chart of the formula related
to this topic.
derive the formula related to this topic.
say the students to solve the problems
by using the formula.
discuss about the process of solving the
problems of perimeter, area and volume
of the solid objects.

Perimeter of plane figures Estimated periods – 3

Adarsha! What do Yes, sir! The closed geometrical
you mean by a figure which is bounded by the
plane figure? straight lines is called the plane
figure. For example triangle,

rectangle, etc.

The total length of the boundary
lines of the given plane figure is

called its perimeter.

Perimeter of triangle

A triangle is a closed figure bounded by three sides
(lines). So, the perimeter of a triangle is the sum of

three sides of it.

Thank you! Adarsha. To find the perimeter of the
triangle, we should measure the sides of the given

triangle with the help of a scale and add all the
measurements.

In the given figure, ABC is a triangle whose three A b cm
sides are AB, BC and CA. B a cm

Sides of DABC Measurement c cm C
AB c cm
BC a cm 53
CA b cm

Now, perimeter of DABC = BC + CA + AB
= a cm + b cm + c cm
= (a + b + c)cm

Prime Mathematics Book - 6

Perimeter of rectangle

A rectangle is a closed figure which is bounded by four sides,
where the opposite sides are equal and each angle is 900. So, the

perimeter of a rectangle is the sum of four sides of it.

The longer side of the rectangle is called its length (l)
and the shorter side of it is called its breadth (b).

In the adjoining figure, ABCD is a rectangle in which D l C
longer and shorter sides are AB and BC respectively. b l b
A B
So, the length = AB = DC = l

and the breadth = BC = AD = b

Now, the perimeter of the rectangle ABCD = AB + BC + DC + AD
=l+b+l+b
= 2l + 2b
= 2(l + b)

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

Perimeter of regular closed figures

Sonu! what do you mean by the Yes, Sir!
regular closed figure?

Please! Give A closed figure in which the lengths
example of each sides are equal is known as

the regular closed figure.

The perimeter of a regular Equilateral triangle,
closed figure is equal to the number square, regular
pentagon, etc.
of sides of the figure times the
measure of a side.

54 Prime Mathematics Book - 6

In an equilateral triangle, all three sides are equal.

So, perimeter of an equilateral triangle a
= No. of sides × measure of a side

= 3 × a = 3a

In a square, all four sides are equal.

So, perimeter of a square = No. of sides × measure of a side

= 4 × a = 4a

In a regular pentagon, all five sides are equal.

So, perimeter of a regular pentagon = 5a a

\ Perimeter of regular polygon = na, where n is the number of sides of the regular
polygon.

Worked Out Examples.

Example 1: Find the perimeter of the given D ABC.

Solution: In the given DABC, AB = 3cm A
5cm
BC = 7cm and AC = 5cm 3cm C
7cm
\ Perimeter of D ABC = AB + BC + CA B

= 3cm + 7cm + 5cm

= 15cm

\ Perimeter of the D ABC is 15 cm.

Example 2: What will be the perimeter of DABC in A
the adjoining figure? Find it. C

Solution: In the adjoining figure, we first measure
all three sides of DABC with the help of
a scale. B

AB = 4cm, BC = 5cm and AC = 3cm

Now, Perimeter of DABC = AB + BC + CA

= 4cm + 5cm + 3cm

= 12cm

Prime Mathematics Book - 6 55

Example 3: Find the perimeter of an equilateral triangle whose one side is 7cm.
Solution: Here, the length of one side of an equilateral triangle (a) = 7cm

Perimeter of equilateral triangle = 3a
= 3 × 7cm
= 21cm

Example 4: Find the perimeter of a rectangle having length 8cm and breadth
5cm.

Solution: Here, length of a rectangle (l) = 8cm
Breadth of the rectangle (b) = 5cm
Now,
perimeter of the rectangle = 2(l + b)
= 2 (8cm + 5cm)
= 2 × 13cm
= 26cm

\ The perimeter of the rectangle is 26cm.

Example 5: If the perimeter of a square is 52cm, find the length of its side.
Solution: Here, perimeter of a square = 52cm

Length of a side (l) =?
Now, perimeter of a square = 4l

or, 52cm = 4l
or, 52 cm = l

4
or, 13cm = l
\ Length of a side of the square (l) = 13cm

56 Prime Mathematics Book - 6

Example 6: Find the perimeter of the given figure. 8cm
Solution: The given figure is

8cm 1cm 5cm
1.5cm 1.5cm

5cm 5cm 5cm

1cm 1cm

1.5cm 1.5cm

Perimeter of the given figure
= 8cm + 5cm + 1.5cm + 1cm + 5cm + 1cm + 1.5cm + 5cm
= 28cm

Example 7: A man runs around a rectangular ground of length 150m and breadth
110m. What distance does he cover in 3 rounds?

Solution: Here,

Length of the rectangular ground (l) = 150m

Breadth of the rectangular ground (b) = 110m

\ Perimeter of the rectangular ground = 2(l + b)

= 2(150m + 110m)

= 2 × 260m

= 520m

Now, the distance covered by the man in 3 rounds

= 3 × 520m

= 1560m

= 1km 560m [ 1km = 1000m]

Prime Mathematics Book - 6 57

Exercise 3.1

1. Find the perimeter of the following figures. (c)
(a) (b) 4cm
2cm 4cm
2cm
4cm 6cm 5cm

(d) (e) 1.5cm 1.5cm (f) 4cm 1.5cm

5cm 3cm 1cm
3.5cm
8cm 7cm
6cm
2. Measure the lengths of the sides of the following figures and find the

perimeter. (b) P (c) W Z
(a) A

B Q RX Y
(d) A C
F
(e) A

ED B

D E

B CC
3. Find the perimeter of an equilateral triangle whose one side is given below.

(a) Length of one side (l) = 6cm (b) Length of one side (l) = 8cm

4. The length (l) and breadth (b) of the rectangle are given below. Find the
perimeter.

(a) l = 10cm and b = 7.5cm (b) l = 12.5cm and b = 9.5cm

(c) l = 4.7cm and b = 2.3cm (d) l = 6cm and b = 3.5cm

5. Find the perimeter of the following regular polygons with given length of
one side.

(a) Square with (l) = 5.6cm (b) Hexagon with (l) = 6cm

(c) Octagon with (l) = 5cm (d) Pentagon with (l) = 7cm

(e) Heptagon with (l) = 4cm

58 Prime Mathematics Book - 6

6. Find the perimeter of the square whose one side is given below:

a) Length of one side (l) = 7cm b) Length of one side (l) = 10cm

7. Find the length of a side of an equilateral triangle with

(a) Perimeter = 21cm (b) Perimeter = 36cm

8. The perimeter and sum of two sides of a triangle are 18cm and 12cm
respectively. Find the length of its remaining side.

9. The perimeter and the length of one side of two equal sides of an isosceles
triangle are 17.2cm and 5.6cm respectively. What is the length of the
remaining side of the triangle?

10. Find the length of a side of a square whose perimeter is

(a) Perimeter = 48cm (b) Perimeter = 28m

11. (a) The perimeter of a rectangle is 60cm and its length is 17cm. Find its
breadth.

(b) The perimeter of a rectangle is 40cm and its breadth is 8cm. Find its
length.

12. (a) A square field is 150m long. If a man is running around it, what distance
does he travel in two rounds?

(b) A rectangular garden is 100m long and 75m broad. What distance will a
boy cover in three complete rounds around it?

13. You are running around a football ground of length 75m and breadth 50m.
How many metres will you cover when you complete 8 rounds around it?

14. A 60m long wire is bent and made a square. Find the length of side of the
square.

Prime Mathematics Book - 6 59

Area of plane figures
Area of regular and irregular shapes

Bablu! Can you say which of the closed figure in each pair
occupy more space?

(a) (b)

(i) (ii) (i) (ii)

In figure (a), the figure (ii) occupy more
space than figure (i) because both are of same shape and size

of (ii) is more. But it is difficult to say in figure (b) because
the figures are of different shape.

Bablu! What about the space occupied by the closed figure
in each of following pairs?

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

In the above figures, I can say easily which figure occupy more space
than other in each pair. Because the same unit and base are used in

each pair of figures.

A closed figure covers a certain portion of a plane. The measure of 1cm
the portion of a plane covered by a closed figure is called the area of 1cm
the closed figure.

The given figure is a square of 1cm × 1cm. So, the surface plane
enclosed by this square is 1 sq.cm

60 Prime Mathematics Book - 6

The area of this square = 1 sq.cm (or 1cm2).

This unit square is used to find the space occupied by any closed figure in the plane
by counting the unit boxes. Area of the closed figure is measured in terms of squares.

How many unit square boxes are in the
adjoining rectangle?

Therefore, area of the There are 10 unit square boxes
rectangle is 10 sq.cm. in the rectangle.

Area of a rectangle

The adjoining figure is a rectangle of length 5cm and breadth 3cm.

Divide the rectangle along the length and breadth into units of length 1cm.

So, along the length it encloses 5 unit square boxes. Similarly, along the breadth it
encloses 3 unit square boxes.

Then, the rectangle is divided into 15 unit square boxes.

\ The area of the rectangle = 15 sq.cm (or 15cm2).

It's area = 15cm2 = 5cm × 3cm = length × breadth

(A) = l × b

Now, length (l) = A = Area
b breadth

And breadth (b) = A = Area
l length

Area of a square 61
A square is a rectangle whose length and breadth are equal.
So, in a square length (l) = breadth (b)
\ Area of the square (A) = length × breadth

= l × l = l2
Thus, area of a square (A) = l2 = (side)2

Prime Mathematics Book - 6

Worked Out Examples.
Example 1: Find the area of the closed figures given below.

(a) (b)

Solution: (a) In this figure,
The number of complete square rooms = 3
The number of half square rooms = 4 = 2 complete square rooms
Thus, total complete squared rooms = 3 + 2 = 5
Therefore, the area of closed figure = 5 sq.cm

(b) In this figure,
The number of complete square rooms = 5
The number of square rooms more than half size = 5
Counting the square rooms more than half size as 1 after
neglecting the square rooms less than half size
The approximate area of the closed figure = 5 + 5 = 10 sq. units.
Therefore, the area of the closed figure = 10cm2.

Example 2: Find the area of a rectangle whose length and breadth are 13cm
Solution: and 9cm respectively.

Here,

Length of the rectangle (l) = 13cm

Breadth of the rectangle (b) = 9cm

Now, area of the rectangle (A) = l × b
= 13cm × 9cm = 117cm2

62 Prime Mathematics Book - 6

Example 3: The area of a rectangle is 24cm2 and its length is 6cm. Find its
Solution: breadth.

Here,
Area of the rectangle (A) = 24cm2

Length of the rectangle (l) = 6cm

Breadth of the rectangle (b) =?

We know that,

b = A = 24cm2
l 6cm
= 4cm

The breadth of the rectangle is 4cm.

Example 4 The area of a rectangle whose length is twice its breadth is 32cm2.
Solution: Find the length and breadth of the rectangle. Also find its perimeter.

Here,
Area of the rectangle (A) = 32cm2

Length = twice its breadth

\ l = 2b

We know that,

A=l×b

or, 32 = 2b × b

or, 2b2 = 32

or, b2 = 32
2
or, b2 = 16

or, b = 16 = 4

\ Length (l) = 2b = 2 × 4 = 8cm

Breadth (b) = 4cm

Again, perimeter of the rectangle = 2(l + b)

= 2(8cm + 4cm)

= 2 × 12cm

= 24cm

Prime Mathematics Book - 6 63

Example 5: The length of a side of a square is 6m. Find its area.
Solution: Here,

Length of a side of the square (l) = 6m
\ Area of the square (A) = l2

= (6m)2
= 6m × 6m
= 36cm2

Example 6: The perimeter and length of a rectangular ground are 120m and
35m respectively. Find the area of the ground.

Solution: Here,

Perimeter of the rectangular ground (P) = 120m

Length of the rectangular ground (l) = 35m

Area of the rectangular ground (A) =?

We know that,

Perimeter of the rectangle = 2(l + b)

or, 120m = 2(35m + b)

or, 120 m = 35 m + b
2

or, 60m - 35m = b

\ b = 25m

Now, area of the rectangular ground (A) = l × b

= 35m × 25m
= 875m2

64 Prime Mathematics Book - 6

Example 7: The area of a square is 121m2. Find its perimeter.
Solution: Here, area of a square (A) = 121m2

Perimeter of the square (P) =?

We know that,
Area of a square (A) = l2

or, 121m2 = l2
l = 121m2 = 11 × 11m2 = 11m

So, the length of a side of the square (l) = 11m

Now, perimeter of the square (P) = 4l

= 4 × 11m = 44m

Example 8: Find the area of the adjoining 2cm 3cm
Solution: figure.
2cm
Here, the given figure is divided 8cm
into different rectangles.

2cm 3cm 12cm

2cm
A C 8cm

B

12cm

In the rectangle 'A', length (l) = 8cm and breadth (b) = 2cm
\ Area of the rectangle 'A' = 8cm × 2cm = 16cm2
In the rectangle 'B', length (l) = 12cm - 2cm - 3cm = 7cm
Breadth (b) = 8cm - 2cm = 6cm
\ Area of the rectangle 'B' = 7cm × 6cm = 42cm2
In the rectangle 'C', length (l) = 8cm and breadth (b) = 3cm
\ Area of the rectangle 'C' = 8cm × 3cm = 24cm2
So, the area of the whole figure = area of part 'A' + area of part 'B'

+ area of part 'C'
= 16cm2 + 42cm2 + 24cm2
= 82cm2

Prime Mathematics Book - 6 65

Example 9: Calculate the area of the shaded 5cm 12cm
region in the figure alongside. 10cm

18cm

Solution: Here,
\
In the given figure, there are two rectangles.

Area of the bigger rectangle (A1) = 18cm × 12cm = 216cm2
Area of the smaller rectangle (A2) = 10cm × 5cm = 50cm2

Now, area of the shaded region is the difference of the area of two
rectangles.

Area of the shaded region = A1 - A2
= 216cm2 - 50cm2

= 166cm2

Exercise 3.2
1. Find the area of every closed figure given below in the graph.

a) b)

2. Find the area of the closed figures given below. (Counting the more than
half square room as 1 and neglecting the squared rooms less than half size.)

a) b)

66 Prime Mathematics Book - 6

3. Find the area of the given figures.

(a) (b) (c)
4cm
5.2cm 3cm

7cm 8.6cm 3cm

(d) (e)
2cm
2cm
9cm
4cm 7cm

10cm 4cm
4cm

4. Find the area of the rectangle whose length and breadth are given below.
(a) Length (l) = 9.4cm, bread (b) = 6.3cm
(b) Length (l) = 12.2m, bread (b) = 8.4m

5. Find the area of the squares whose one side is given.

(a) Side (l) = 7cm (b) Side (l) = 15cm (c) Side (l) = 8.4m

6. Find the measure of unknown side of each given closed figures.

(a) (b) A = 60cm2 (c) (d)
? A64=cm2 ?
8cm
A = 32cm2 ? 6cm 3cm
A = 36cm2
8cm ?

7. (a) The measure of one side of a square is 12cm. Find its area and perimeter.

(b) The area of a square is 169m2. Find the measure of its side and also its
perimeter.

(c) The area of a rectangle is 72m2 and its length is 12m. Find its breadth
and perimeter.

Prime Mathematics Book - 6 67

8. (a) The area of a rectangle whose length is three times its breadth is 75m2.
Find the length and breadth of the rectangle.

(b) The area of a rectangular room is 72m2. If its breadth is half of its length,
find its perimeter.

9. (a) The perimeter and breadth of a rectangle are 40cm and 8cm respectively.
Find the area of the rectangle.

(b) The perimeter and length of a rectangular bed are 22 feet and 7 feet
respectively. Find the area of the bed.

(c) The perimeter of a square field is 500m. Find the area of the field.

10. Find the area of the shaded region in the given figures.

(a) 3cm (b) (c)
7cm
4cm 2cm
1cm 2cm

12cm
6cm
4cm
6cm

15cm 8cm

(d) 8cm(f) 7cm
3cm 4cm
2cm
10cm 7cm

68 Prime Mathematics Book - 6

Volume of Solids

The space occupied by a solid is called its volume. The 1cm
volume of the solid is measured in cubic millimeter 1cm
(cu. mm or mm3), cubic centimeter (cu. cm or cm3), cubic 1cm
metre (cu. m or m3) etc.

The adjoining figure is a cube of a block. Its length = 1cm, breadth = 1cm and
height = 1cm. So, its volume is 1 cu. cm.

Volume of cuboid 2cm

The adjoining block contains 12 small 3cm 2cm
blocks where the volume of each small
block is 1 cu.cm.

So, volume of adjoining block is 12 cu.cm.

Now, measure the length, breadth and height of the given block.

Its length = 3cm, breadth = 2cm and height = 2cm

Volume of the cuboid = length × breadth × height

= 3cm × 2cm × 2cm

= 12 cu.cm

Volume of the cuboid = length × breadth × height

=l×b×h
Volume of cube
In a cube, length, breadth and height all are equal
Length (l) = breadth (b) = height (h)
Volume of the cube = length × breadth × height

=l×b×h
=l×l×l
= l3

Prime Mathematics Book - 6 69

Worked Out Examples.
Example 1: A brick is 10cm long, 6cm broad and 5cm in height. Find its volume.
Solution: Here,

Length of brick (l) = 10cm
Breadth of brick (b) = 6cm
Height of brick (h) = 5cm
\ Volume of brick (V) = l × b × h

= 10cm × 6cm × 5cm = 300 cu.cm

Example 2: Find the volume of a cube of length 8cm.
Solution:
Here,

Length of a cube (l) = 8cm = 512 cu.cm
\ Volume of the cube (V) = l3

= (8cm)3

Example 3: The length and breadth of a cuboid are 15cm and 9cm respectively.
Solution:
If the volume of the cuboid is 810 cu.cm, find its height.

Here,

Length of a cuboid (l) = 15cm

Breadth of a cuboid (b) = 9cm

Volume of the cuboid (V) = 810 cu.cm

Height of the cuboid (h) = ?

Now, volume of the cuboid (V) = l × b × h

or, 810 cu.cm = 15cm × 9cm × h

or, 810 cu.cm = 135 sq.cm × h

or, h = 810 cu.cm
135 sq. cm

\ h = 6cm

\ The height of the cuboid is 6cm.

70 Prime Mathematics Book - 6

Example 4: The length of a rectangular block is twice its breadth and its height
Solution: is 5cm. If the volume of the block is 640 cu.cm., find its length and
breadth.
Here,
Length of the rectangular block (l) = 2 × breadth = 2b
Height of the block (h) = 5cm
Volume of the block (V) = 640 cu.cm
Now, volume of the block (V) = l × b × h

or, 640 cu.cm = 2b × b × 5cm
or, 640 cu.cm = 2b2 × 5cm
or, b2 = 640 cm3

2 × 5cm
or, b2 = 64 cm2
or, b = 64 cm2 = 8cm

So, the breadth (b) = 8cm
\ Length (l) = 2b = 2 × 8cm = 16cm

Exercise 3.3

1. Find the volume of the given solids. 1cm
(a) (b) 1cm
5cm 3cm

4cm (d)
8cm
2cm
(c) 3cm
4cm

4cm 6cm
4cm

Prime Mathematics Book - 6 71

2. Find the volume of the cuboids with the following measures.
(a) length (l) = 4cm, breadth (b) = 2cm and height (h) = 5cm

(b) length (l) = 12cm, breadth (b) = 8cm and height (h) = 6cm

3. Find the volume of the cubes with the following measure.

(a) Length of a side (l) = 5cm (b) Length of a side (l) = 9cm

4. A block is 18cm long, 14cm broad and 10cm high. Find its volume.
5. Find the volume of a cube having the length of an edge 6cm.
6. What will be the length of a cubical box having its volume 512 cu.cm?
7. A water tank is 1m long, 70cm broad and 50cm high.

(i) Find the volume of the tank.
(ii) If the tank is completely filled with water, what is the volume of water?
(iii) If 1000 cu.cm = 1 litre, how many litres of water does it hold?

8. The length of a cuboid is twice its breadth and its height is 5cm. if the
volume of the cuboid is 250 cu.cm, find its length and breadth.

9. The height of a box is half of its breadth and its length is 18cm. If the
volume of the box is 1296 cu.cm, find its breadth and height.

10. The volume of a rectangular box 8cm long, 4cm broad and 2cm high is equal
to the volume of a cubical box. Find the length of one side of the cubical
box.

72 Prime Mathematics Book - 6

Unit Revision Test I

1. a) Find the perimeter of a triangle having the sides 4cm, 6cm and 8cm.
b) Find the perimeter of a rectangle having length 12cm and bredth 8cm.

c) A boy runs around a rectangular ground of length 120 m and bredth
80 m. What distance does he cover in 5 rounds?

2. a) Find the area of a rectangle, whose length and breadth, are 15cm and
8 cm respectively.

b) The area of a rectangle whose length is twice its breadth is 72 cm2. Find
the length and breadth of the rectangle. Also find its perimeter.

c) The perimeter and breadth of a rectangle are 50 cm and 10 cm respectively.
Find its area.

3. Find the area of the shaded region in the given figures.
a) b)

6 cm 14 cm 2 cm
1.5 cm
9 cm 2.5 cm
18 cm
3.5cm 1.5cm 3.5cm

4. a) A book is 15 cm long 10 cm broad and 4 cm thick. Find its volume.

b) Find the volume of a cube of length 12 cm.

c) The length and bredth of a cuboid are 16 cm and 10 cm respectively. If
the volume of the cuboid is 800 cu.cm, find its height.

d) The length of a cuboid is twice its breadth and its height is 7 cm. If the
volume of the cuboid is 896 cu.cm find its length and breadth.

Prime Mathematics Book - 6 73

Estimated periods – 9

Objectives

At the end of this unit, the students will be able to:
understand transformation, isometric and non-isometric transformation.
invariant point and identify transformation.
translate and reflect given figures in simple plane.
identify the symmetrical and nonsymmetrical figures.
draw axis/axes of symmetry of the given symmetrical figures.
get introduction of tessellation and draw simple tessellated designs.
construct attractive designs of polygons and polygrams.

Al

P A'
CR
C'

B
Q
B'

Teaching Materials Activities

Plane mirror, set squares, magnifying glass. It is better to:
different symmetrical figures, attractive show reflection on plane mirror, show
designs of polygons and polygrams, formation of image by magnifying glass,
designs of tessellation, models of slide set square along the age of scale to
polygons. clarify the idea of transformation.
discuss about isometric and non-isometric
transformation and basic transformation.
demonstrate paper models of
symmetrical shapes and show the axes
of symmetry and discuss about number
of axes of symmetry.
demonstrate designs of polygons.
ask students to tessellate given polygon.

Transformation Estimated periods – 4

Literal meaning: transformation = change

In geometry, making changes in points, lines and figures under certain rule is called
transformation. Changes may be in position, size and shape.

Observe the following transformation

1.

Reduced photo

Enlarged photo

2. screen

Lens

Image focused on a screen
3.

Reflected image (mirror image) 75

Prime Mathematics Book - 6

4. A' D'

AD
B' C'

BC
Figures moved through a certain distance along certain direction.
5.

Figure rotated about a point

Under a transformation, let the position of a point P be changed to P', the point P'
is known as the image of the point P, mathematically we say, under transformation
T, the point P maps to P'. Symbolically we write T:(P) ® P'

In the figure given alongside, rotating the part ABCD through 90o about the point O,
we get its image A'B'C'D' forming the symbol swastika where.

A ® A' A' B A
B ® B' C'
C ® C' O' D'
D ® D' B' O

DC

The point O remains unchanged. Such points are called invariant points.

Thus, the points which remain unchanged under a transformation are called invariant
points. Sometimes a figure remains unchanged under a transformation i.e. image
of a figure coincide with the figure. Such transformation is called

76 Prime Mathematics Book - 6

indentity transformation. In the figure B A A' C'
given alongside, DABC is transformed C
to DA'B'C' . The figure and its image C B'
are congruent (equal and similar). Such Isometric transformation
transformation is called an isometric
transformation.

In the figure given alongside, DABC A
® DA'B'C'. The figure and its image
A

are similar but not congruent

(different size). Such transformation B CB

is called non-isometric non-isometric transformation

transformation.

Basic transformations: There are different types of transformations. But basic
transformations are as follows.

Basic isometric transformations:

(a) Translation (b) Reflection (c) Rotation

Basic non-isometric transformation: Enlargement or reduction. Here, we will discuss
only about translation and reflection in simple plane.

Translation: In translation, a point or figure is displaced by certain distance and

direction. It is an isometric transformation. Since each and every point get displaced,

there exist no invariant points.

Q

To translate DABC by given vector PQ. A'

- From each vertex A, B, C, draw lines parallel P
to PQ along the direction of PQ.

- Take AA' = BB' = CC' = PQ. A C'
- Join A'B', B'C', A'C' B'

Thus, TPQ : DABC ® DA'B'C' B C

Reflection: Under reflection on a line l, a point P is P l
mapped to P' such that the line l is perpendicular P'
bisector of PP'. The line l is called the line (axis) of
reflection.

We write, Rl : P ® P'

Reflection is an isometric transformation

Prime Mathematics Book - 6 77

To reflect DABC on the line l: A l
B
- Given DABC and axis l, draw AP, BQ, CR P A'
perpendicular to l. CR
C'
- Produce AP, BQ and CR to A', B' and C'
respectively such that AP = PA' Q
B'
BQ = B'Q and CR = RC'

- Join A'B', A'C', B'C'

Thus, Rl: DABC ® DA'B'C'

Exercise 4.1

1. Define: (b) Invariant point
(a) Transformation (d) Isometric transformation
(c) Identity transformation

2. Name the basic (fundamental) transformations and say which of them are
isometric?

3. Write the following symbols in words.

(a) TA®B: P ® P' (b) TP®Q: DABC ® DA'B'C'
(c) Rl: P ® P' (d) RPQ: DABC ® DA'B'C'

4. Mention the change in position or size or both in the following when the
figure A maps to the figure B.

(a) (b)

A BB A

(c) (d) B
A BA

78 Prime Mathematics Book - 6

5. Copy the following figures approximately and translate by the given directed
line segments.

(a) A P Q
C

B D (c) F
(b) A

A

ED

BC BC

6. Copy the following figures approximately and reflect them on the given
line.

(a) A (b) A

C B

B C
(c) (d)

Prime Mathematics Book - 6 79

Symmetry, Designs of Polygons and Tessellation Estimated periods – 5

Symmetrical figures X

Spray few drops of ink on the paper and fold the paper
along a line XY before the ink get dried. You will find
a figure (pattern) which is identical about the line.

Observe these figures. Y

Left and right parts of each of these figure are identical. Such features of figures
are called symmetry. The straight line about which a figure is symmetrical is called
axis (or line) of symmetry. A symmetrical figure may have one or more axes (lines)
of symmetry.

1.

These figures have only one axis (line) of symmetry.
2.

These figures have two axes (lines) of symmetry.
3.

These figures have three axes (lines) of symmetry.

80 Prime Mathematics Book - 6

4.

5

A square has 4 A hexagram has A circle has infinite
four axes of six axes of number of axes of
symmetry 123 symmetry
symmetry
A pentagon has 5
axes of symmetry

Note: If a figure is symmetrical about a straight line, the symmetry is called bilateral (plane
or reflection) symmetry.

These figures or symbols seem to be symmetrical but they are not bilaterally
symmetrical. Such symmetry is called rotational (radial) symmetry.

Exercise 4.2

1. Use ruler to draw possible axes of symmetry of the following letters.

ABCDEHIK
MOTUVWXY

2. Draw all possible axes of symmetry of the following geometrical figures.

Prime Mathematics Book - 6 81

3. The dotted lines represent the axes of symmetry. Complete the following
figures.

4. Write the possible number of axes of symmetry of the following figures.

5. Write the possible number of axes of symmetry.

(a) Capital letter A. (b) Capital letter F.

(c) Capital letter X. (d) Capital letter M.

(e) An equilateral triangle. (f) A rectangle

(g) A circle (h) A pentagram

6. (a) Write the letters of English alphabets (in capital) which are not symmetrical.
(b) Draw any two figures with a horizontal axis of symmetry.
(c) Draw any two figures with a vertical axis of symmetry.
(d) Draw any two figures with two axes of symmetry.
(e) Draw any two figures with three axes of symmetry.
(f) Draw any two figures with four axes of symmetry.

82 Prime Mathematics Book - 6

Designs of Polygons

By constructing smaller polygons inside a polygon, we can get attractive patterns.
If we colour the polygons in regular way, we can get an attractive and decorative
designs.

Designs from an equilateral triangle: - Construct an equilateral
1. triangle.

2. - Join the mid points to form
inner equilateral triangles.

- Colour the triangles in regular
way.

- Construct an equilateral triangle
- Join the mid points to form inner equilateral triangles.
- Join vertices to mid points of opposite sides.
- Colour the triangles alternately.

Designs from square:
1.

2.

3.

Prime Mathematics Book - 6 83

Design from regular pentagons
1.

2. 72o

Designs of regular hexagon
1.

60o

2.

60o

Other patterns 5 6 7 8 9 10 11 12 13 - Draw any angle
1. 5 6 7 8 9 10 11 12 13 -
- Divide the arms into equal intervals
1234 as shown.
1234 -
Join 1st of one and last of next,
2nd of one and 2nd last of next, 3rd
of one and 3rd last of next and so
on.

Name the curve so described.

84 Prime Mathematics Book - 6

2.

- Take two points
- Draw concentric circles as in the figure
- Colour as shown
- The figure described by the colour pattern are called ellipses

Tessellation

In Latin, tessela means small cubic pieces of clay, stone or glass used to make mosaics
the art of creating decorative patterns of imaged on walls, ceilings and floors.

Honey comb Tiling Paved road Bamboo knitting

Tessellation is the process of creating pattern on a plane using repeatition of
geometric shape with no overlaps and no gaps.

Prime Mathematics Book - 6 85

Historical fact
Johannes Keplar (1571 - 1630, Germany) first made documented
study of tessellation in 1619.

Honey comb is an example of tessellated natural structure. For tiling the walls,
paving the roads, glazing goods with patterns, providing designs in textiles, bamboo
and other knittings, idea of tessellation is used.
Taking a tessela or a block on a plane different patterns can be formed by putting
other tessela or blocks in translated or reflected position about the first.
Observe the following tessellations.

86 Prime Mathematics Book - 6

Exercise 4.3

1. Draw an equilateral triangle
of side 8cm and complete
the pattern as shown.

2. Draw a square of side 8cm and
complete the pattern.

3. Draw a regular 72o
pentagon (taking
a circle of radius
6cm) and draw
the design.

Prime Mathematics Book - 6 87

4. Draw a regular 60o
hexagon (taking a
circle with radius
6.4cm) and
complete the
pattern as shown.

5. Complete the following tessellations (use grid or graph).
(a) (b)

(c)

(d)

88 Prime Mathematics Book - 6

Unit Revision Test
1. Define: (a) Invariant point (b) Isometric transformation
2. Name the basic transformation which are isometric.
3. Copy the given figure approximately and translate by the given directed

line segment.

4. Copy the given figure approximately
and reflect it on the given line (l).
5. Complete the following
tessellation.
l

6. Copy the given figures approximately and draw possible axes of symmetry.
(a) (b) (c)

Prime Mathematics Book - 6 89

Estimated periods – 15

Objectives

At the end of this unit, the students will be able to:
define a set
represent the sets in different ways
classify the sets on the basis of cardinalily

A set of wild animals.

Teaching Materials Activities

Charts of different objects avaliable in It is better to:
school, flash cards, number cards ask the students collect differnt objects
of same categorie, name the sets.
help the students to represent the sets in
different ways.
let the students count the elements of
the sets and classify them.

Introduction

It is a collection of fruits. So, it is a set of fruits.
Could you say whether the cauli-flower belongs
to the given set or not? It is clear that cauli-
flower is not a fruit, so it doesn't belong to the
given set.

Does apple belong to the set? Yes, it belongs to
the set because apple is a fruit. Thus, a set should
be well defined.

Is this a collection of tall girls ?
No, it's not a set because tall girl is not well-
defined.
Thus, a set is a collection of well defined distinct
objects.

Elements of a set It is a set of wild animals.

What are the elements of the given set? The
elements of the given set are tiger, bear, lion
and elephant. Thus, the objects belonging to the
set are called elements of the set. Cow doesn't
belong to the given set so cow is not a member
of the given set.

Set notation
A = {a, e, I, o, u}

It is a set of vowels. a, e, i, o and u are the elements of a set A. The elements a,
e, i, o and u are separated by comma and enclosed with brackets. A set is denoted
by capital letters A, B, C etc.

Again, consider

B = {potato, tomato, cabbage, tomato, potato, brinjal}

In the set B, the elements are repeated. It should not happen. That means elements
should not be repeated.

Prime Mathematics Book - 6 91

\ B = {potato, tomato, cabbage, brinjal}

Î ÎPotato is a member of the set B. It is denoted by potato B. (epsilon) means

"is a member of " or " belong to".

ÎOrange doesn't belong to the set B. It is represented by orange B. Means "is not

a member of" or "doesn't belong to".

Exercise 5.1

1. Find out which collection are well defined.
(a) The set of fat boys of class.
(b) The collection of first 10 natural numbers.
(c) The collection of strong students of class VI.
(d) The collection of odd numbers less than 30.
(e) The set of days of the week.

2. Write the elements of the following sets, naming the sets.
(a) The set of days of the week.
(b) The set of first five prime numbers.
(c) The set of English months of the year.
(d) A set of five fruits.
(e) The set of letters of the word "Mathematics".
(f) The set of vowels of English alphabet.
(g) The set of four mountains of Nepal.
(h) The set of first four letters of Devenagari script.
(i) The set of four colours.

Î Ï3. Use the correct symbol or in the gaps.

(a) 1 ………………. {set of prime numbers}

(b) 3 ………………. {set of odd numbers}

(c) Rose ………………. {set of flowers}

(d) Apple ………………. {set of vegetables}

(e) 4 ………………. {set of factors of 12}

(f) 5 ………………. Set of prime factors of 20}

(g) 2 ………………. {set of even numbers}

(h) a ………………. {set of consonants of English alphabet}

(i) 2 ………………. {set of composite numbers}

(j) 4 ………………. {set of cubed numbers}

92 Prime Mathematics Book - 6

Representation of a set

Usually sets are represented by the following methods.

1. Description method 2. Tabulation method or Roaster method

3. Set-builder form or Rule method

1. Description method:

In this method, sets are described in a phrase or sentence and kept within
brackets. The description is made in such a way that it fully describes the
properties of the elements.

Examples:
A = {First five natural numbers.}
B = {Twelve months of Nepali calendar}
C = {Planets in the solar system}

2. Roaster method:

D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

E = {2, 4, 6, 8, 10}
F = {1, 2, 3, 4, 5}
In above examples, the elements of the set are listed within the braces.
Thus, in Roaster method, the elements are listed and separated by commas
within the brackets.
3. Set-builder method:

G = {x: x is a factor of 12}

This is read as "G is a set of all x such that x is a factor of 12.
H = {x: x is a natural number}

H is a set of all x such that x is a natural number.
I = {x / x is an even number}
I is a set of all x such that x is an even number.
The symbol (:) or (/) stands for such that.
Thus, in this method, a rule or formula is written within the braces that defines
the set.
4. Diagramatic method: If sets are shown in diagram.

Note: 1. If the number of elements in a set is finite we use roaster method.
2. For example: A = {prime numbers less than 10}
If the number of elements in a set is infinite, we use rule method.
For example: B = {x : x is a prime number}

Prime Mathematics Book - 6 93

Exercise 5.2

1. List the elements of the following sets in Roaster method.

(a) A set of five fruits.
(b) The set of first three months of English calendar.
(c) The set of English days starting with s.
(d) The set of first five natural numbers.
(e) The set of all countries of SAARC.
(f) A set of four colours.
(g) The set of letter of the word "Mathematics".
(h) A set of vowels in English alphabets.
(i) The set of first four square numbers.
(j) The set of first five multiples of 5.

2. Describe the following sets in words.

(a) A = {2, 3, 5, 7, 11} (b) B = {1, 2, 3, 4}

(c) C = {2, 4, 6} (d) D = {1, 4, 9, 16, 25}

(e) E = {1, 8, 27, 64, 125}

(f) F = {Baisakh, Jestha, Asadh, Shrawan, Bhadra}

(g) G = {Mango, Orange, Banana, Apple}

(h) H = {Red, Black, Green, White}

(i) I = {Cow, Cat, Dog, Goat}

(j) J = {Potato, Tomato, Cabbage, Brinjal}

3. Write the following sets in set-builder method.

(a) A = {January, February, March, April }

(b) B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

(c) C = {1, 3, 5, 7} (d) D = {2, 4, 6, 8, 10}

(e) E = {2, 3, 5, 7, 11} (f) F = {1, 4, 9, 25, 36}

(g) G = {1, 8, 27, 64, 125} (h) H = {-2, -1, 0, 1, 2}

(i) I = {a, e, i, o, u} (j) (J) {cow, cat, dog, sheep}

94 Prime Mathematics Book - 6