10

10. Mensuration

ÒPERIMETER Ò

Introduction : The concept of perimeter is used in our daily life. Consider these
situations :

1. Mr. Saxena wants to construct a boundary wall around his house.

2. Dr. Sinha does his morning walk along the walking track built around the local park.

Here, we should find the perimeter of the given figure.

Perimeter is the distance covered along the boundary of a close figure. In this
chapter, we will learn about the perimeter of some closed figure.

I. Perimeter of Regular-shaped Figures : A simple closed figure is said to be of

regular shape, if its all sides are equal. Following are some regular-shaped figures :

bc ef

d de ef f

aa bc c
b

a b c dd ee ff
An Equilateral triangle A Rhombus A Square e f
d
A Pentagon A Hexagon An Octagon

Perimeter of Equilateral Triangle

Perimeter = 3 × length of one side

Perimeter of Square or Rhombus

Perimeter = 4 × length of one side

Perimeter of Regular Pentagon

Perimeter = 5 × length of one side

Similarly, we have the following :

Perimeter of a regular hexagon = 6 × length of one side

Perimeter of a regular octagon = 8 × length of one side

Therefore, the perimeter of a ‘n’ sided polygon is

= n (no. of sides) × length of one of its side. l

Perimeter of a Rectangle : Sum of its all sides A B
b
= AB + BC + CD + DA b C

P=l+b+l+b

Dl

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P = 2l + 2b

Perimeter = 2 (length + breadth)

(l denotes – its length, b denotes – its breadth
and p denotes – its perimeter)

Example : A tabletop measures 2 m 50 cm by 1 m 50 cm. What is the perimeter of the
tabletop?

Solution : Length of the tabletop = 2 m 50 cm = 2.50 m

Breadth of the tabletop = 1 m 50 cm = 1.50 m

So, Perimeter of the tabletop = 2 (length + breadth)

= 2 (2.50 + 1.50) m = 2 (4) m = 8 m

II. Perimeter of Irregular Figures : If the sides of a closed figure are not of same

measure, then the figure is said to be irregular. The perimeter of a polygon is the sum of the

length of all its sides. For example : The perimeter of the polygon pictured below is equal to

1 + 5 + 4 + 2 + 7 = 19. 4

2 It should be kept in mind that if the Q
length of the sides of the polygon are U
5 I
C
in cm then the perimeter will also be in K

7 cm.

1 TIP

Short cut formulae :

Side of the square = P(Perimeter of square) units
4

Length of Rectangle = P(Perimeter of rectangle) – b
2

Breadth of Rectangle = P(Perimeter of rectangle) – l
2

ÒAREA Ò

Introduction : We know that a closed figure encloses some region in it. Area is the
surface or region enclosed in a closed plane figure.

Standard units of area are cm2, m2, km2, hectare.
1 sq. cm. = 1cm2

Area of Rectangle :

Area of rectangle = length of Area × breadth of Area

or Area = l × b (l denotes – its length, b denotes – its breadth)

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Example : Find the area of the rectangle whose sides are 6 cm and 4 cm.

Solution : Suppose that –

Length of the rectangle = 6 cm

Breadth of the rectangle = 4 cm

Area of rectangle = length × breadth
= 6 cm × 4 cm = 24 cm2

Short cut formulae :

length = Area of Rectangle
Breadth

breath = Area of Rectangle
Length

Area of Square :

Area of square = side × side
Area of square= side2

Example : Find the area of the square whose side is 5 cm.

Solution : Side of square = 5 cm

So, Area of square = side × side
= 5 cm × 5 cm = 25 cm2

Advance Tip

Some Important Conversions :

Unit of length Units of area

1 cm = 10 mm 1 sq. cm = 100 sq. mm

1 m = 100 cm 1 sq. m = 10,000 sq. cm

1 dm = 10 cm 1 sq. dm = 100 sq. cm

1 dam = 10 m 1 sq. dam = 100 sq. m

1 km = 1,000 m 1 sq. km = 10,00,000 sq. m

1 Are = 100 m2 and 1 Hectare = 10,000 m2

Advance Tip
A square has a larger area than all other quadrilaterals with the same perimeter.

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W arm Up

A. Choose the correct option from the four options given below. Mark your choice in the
answer sheet printed at the end of the questions.

1. Perimeter is always measured in :

(a) sq.m (b) sq.cm (c) cm (d) given units

2. Area is always measured in :

(a) given units (b) cm (c) m (d) sq. units

3. Area of a rectangle is 240 m2. If its length is 16 m, then its breadth is :

(a) 15 m (b) 8 m (c) 32 m2 (d) 64 m

4. Area of a square park whose side is 8 m is :

(a) 32 m (b) 64 m2 (c) 32 m2 (d) 14 m
?
5. The perimeter of the given figure is 23 cm. 4 cm

Find the missing value : 3 cm

(a) 6 cm (b) 4 cm

(c) 7 cm (d) 5 cm 11 cm

6. If the length and breadth of a rectangle are doubled, then its perimeter is :

(a) halved (b) doubled (c) tripled (d) none of these

7. Perimeter of a rectangle whose length is ‘l ’ and breadth is ‘b’ is given by :

(a) 2(l + b) (b) 2 l + b (c) 2 b + l (d) 2 (l × b)

8. The perimeter of a rhombus is ____________ times the length of the side.

(a) 2 (b) 4 (c) 3 (d) 5

9. Two squares of side 4 cm each are joined together. The perimeter of the resulting
figure is :

(a) 16 m (b) 32 cm

(c) 24 cm (d) 64 cm

10. The area of a square is 225 cm2. The perimeter of square will be :

(a) 15 cm (b) 25 cm (c) 45 cm (d) 60 cm

Practice Session

11. The perimeter of regular polygon is : (b) no. of sides + length of one side
(a) no. of sides × length of one side (d) no. of sides ÷ length of one side
(c) no. of sides – length of one side
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12. If the area of rectangle increases from 2 cm2 to 4 cm2, the perimeter will :

(a) increase (b) decrease (c) remains same (d) none of these

13. Which figure encloses more area : a square of side 2.5 cm; a rectangle of sides
3 cm and 2 cm ?

(a) rectangle (b) square (c) same (d) can’t compare

14. The area of rectangle whose length is 15 cm & breadth is 6 cm :

(a) 9000 cm2 (b) 90 cm2 (c) 9 cm2 (d) 900 cm2

15. The perimeter of square whose side is 2.5 m is :

(a) 10 m (b) 10 m2 (c) 6.25 m2 (d) 6.25 m

16. The perimeter of rectangle with length 2 cm and breadth 1.5 cm is :

(a) 3.4 cm (b) 7 cm (c) 6 cm (d) 3.5 cm

17. The area of a rectangular carpet is Q
U
120 m2 and its perimeter is 46 m. l + b = 23 and lb = 120. I

The length of its diagonal is : (l2 + b2 ) = (l + b)2 – 2lb =(23)2 -2 ´120 =289. C
K
(a) 15 m (b) 16 m Diagonal =l2 +b2 =289

(c) 17 m (d) 20 m TIP

18. The length of a rectangle is three times its width and the length of its diagonal is
6 10 cm. The perimeter of the rectangle is :

(a) 48 cm (b) 36 cm (c) 24 cm (d) 24 10 cm

19. If the ratio between the length and perimeter of a rectangular plot is 1 : 3, then

the ratio between the length and breadth of the plot is :

(a) 1 : 2 Q
(b) 2 : 1 Let the length be x cm. Then, its perimeter U
(c) 3 : 2 I
(d) 2 : 3 is 3x cm. C

\2(x + b) = 3x Þ2b = (3x - 2x) = x Þb = x . K
2

TIP

20. How many envelopes can be made out of a sheet of paper 72 cm by 48 cm, if

each envelope requires a paper of size 18 cm by 12 cm?

(a) 4 (b) 8 (c) 12 (d) 16

ADVANCELEVEL

21. Adjusting figure is made up of an equilateral triangle and a square of
side 7 cm. The perimeter of the figure is :

(a) 42 cm (b) 35 cm (c) 32 cm (d) 49 cm

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22. In the given fig., ABCD is a square and PQR is an D CP
equilateral triangle. The difference between the
perimeters of these two geometrical shapes is : 4.5 cm 4.5 cm

(a) 0 (b) 13.5 cm A BQ R
(c) 5 cm (d) 4.5 cm

23. Perimeter of a square is equal to that of an equilateral triangle. If perimeter of the
triangle is 21.6 cm, side of the square is :

(a) 7 cm (b) 5 cm (c) 5.4 cm (d) 6 cm

24. A square flower-bed is surrounded by a path 10 m wide around it. If the area of
the path is 2000 m2, the area of the square flower-bed will be :

(a) 2100 m2 (b) 1600 m2 (c) 1440 m2 (d) 1200 m2

25. A lane 150 m long and 9 m wide is to be paved with bricks, each measuring 22.5
cm × 7.5 cm. How many bricks are required?

(a) 65000 (b) 70000 (c) 75000 (d) 80000

26. If side of a square is x cm, its area is given by :

(a) 2x sq. cm (b) x2 sq. cm (c) 4x sq. cm (d) x sq. cm

27. If x and y are length and breadth of a rectangle respectively, its area is given by :

(a) xy sq. unit (b) (c) 2xy sq. unit (d) x + y sq. unit.
28. If length and breadth of ayxrseqc.tuannigt le are doubled and halved respectively, its area

becomes :

(a) two times (b) four times (c) unchanged (d) half

29. Area of the region as shown in the given fig. is :

2 (a) 28 sq. m
3

2 (b) 26 sq. m
24 (c) 24 sq. m

44 (d) 25 sq. m
3
Q
3 2 2 A figure can be split into more than one U
1 44 ways. I
2 The figure has been split into 4 rectangles C
2 K
2
3 Area of the rectangle 1 = 4 × 2 = 8 sq. unit
2
Area of the rectangle 2 = 6 × 1 = 6 sq. unit TIP

41 2 Area of the rectangle 3 = 3 × 2 = 6 sq. unit
3
Area of the rectangle 4 = 4 × 2 = 8 sq. unit

21
1

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30. Length and breadth of a rectangle are 48 cm and 12.5 cm respectively. If the
length is increased by 2 cm and breadth is increased 2 times, then area of the
new rectangle so formed is :

(a) 1200 sq. cm (b) 1000 sq. cm (c) 1150 sq. cm (d) 1250 sq. cm

31. The perimeter of a rectangle is 32 cm. Which of the following cannot be its
dimensions?

(a) 15 cm, 1 cm (b) 8 cm, 8 cm (c) 11 cm, 5 cm (d) 14 cm, 3 cm

32. The length of one side of a square is more than 4.2 cm. Which of the following
can be its perimeter?

(a) 16.4 cm (b) 16 cm (c) 17.2 cm (d) 16.2 cm

33. Three squares of side 4 cm each are joined together. The perimeter of the
resulting figure is :

(a) 16 cm (b) 32 cm (c) 24 cm (d) 64 cm

1. a b c d 2. a b c d 3. a b c d 4. a bc d
5. a b c d 6. a b c d 7. a b c d 8. a bc d
9. a b c d 10. a b c d 11. a b c d 12. a bc d
13. a b c d 14. a b c d 15. a b c d 16. a bc d
17. a b c d 18. a b c d 19. a b c d 20. a bc d
21. a b c d 22. a b c d 23. a b c d 24. a bc d
25. a b c d 26. a b c d 27. a b c d 28. a bc d
29. a b c d 30. a b c d 31. a b c d 32. a bc d
33. a b c d

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