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### Important Terms Related to Mensuration Formula

Before understanding the mensuration formulas, we need to understand certain terms. These are –

● Perimeter: This is measured in units such as m, cm, etc and it is the measure of or sum of the continuous length of the boundary of a figure.

● Area: This is measured in square units such as m², cm², etc and it is the surface enclosed in a figure.  (Mensuration Formula PDF Download)

● Volume: This is measured in cubic units such as m³, cm³, etc, and is nothing but the space occupied by an object.

● Curved/Lateral Surface area: This is measured in square units such as m², cm², etc and it is the area of the curved surface in a figure.

● Total Surface area: This is measured in square units such as m², cm², etc and it is the area of the total surface in a figure including the top and bottom portions.

#### Mensuration Questions with Solutions –

Q1. The perimeters of two similar triangles ABC and PQR are 156 cm and 46.8 cm respectively. If BC= 19.5 cm and QR= x cm, then the value of x is?

Solution: (△ABC) Perimeter/ (△PQR) Perimeter) = BC/QR

=> 15.6/46.8 = 19.5/x

=> x = 46.8 x 19.5 / 156

Q2. A solid metallic sphere of radius 10 cm is melted and recast into spheres of radius 2 cm each. How many such spheres can be made?

Solution: As, V = 4/3 π r³

Volume of big sphere = n x volume of small sphere

=> 4/3 π (10)³ = n x 4/3 π (2)³

=> n = 125

Q3. A circle has a radius of 21 cm. Find its circumference and area. (Use π = 22/7)

Solution: We know, Circumference of circle = 2πr = 2 x (22/7) x 21 = 2 x 22 x 3 = 132 cm

Area of circle = πr² = (22/7) x (21)² = 22/7 x 21 x 21 = 22 x 3 x 21

Q4. The length, width, height of a cuboidal box are 30 cm, 25 cm and 20 cm, respectively. Find its area.

Solution: Here, l- 30 cm, b= 25 cm, and h= 20 cm

Total surface area = 2 (lb +bh +hl)

= 2 (30 × 25 + 25 × 20 + 20 × 35)

TSA = 2( 750 + 500 + 700) = 3900 cm²

Q5. A rectangular piece of paper 11 cm × 4 cm is folded without overlapping to make a cylinder of the height of 4 cm. Find the volume of the cylinder.

Solution: The length of the paper will be the perimeter of the base of the cylinder and the width will be its height.

Circumference of base of cylinder = 2πr = 11 cm

2 x 22/7 x r = 11 cm (Mensuration Formula PDF Download)

r = 7/4 cm

Volume of cylinder = πr²h = (22/7) x (7/4)² x 4 = 38.5 cm³

Q7. The total surface area of a hemisphere is 166.32 cm², find its curved surface area?

Solution:  Let radio of hemisphere = r cm

Tital surface area of hemisphere = 3 π r² = 166.32 ……………….. (i)

Multiplying equation (i) by ⅔

=> ⅔ x 3 π r² = ⅔ x 166.32

=> Curved surface area of hemisphere = 2 π r² = 2 x 55.44 = 110.88 cm²

Q8. If G is the centroid and AD, BE, and CF are three medians of the triangle with 72 cm², then the area of triangle BDG is:

Solution: The area of the triangle formed by any two vertices and centroid is ⅓) times the area of △ABC. (Mensuration Formula PDF Download)

Also, the median divides the triangle into two equal areas.

So, the area of △BDG = 1/6 times of △ABC

=> 1/6 x 72

=> 12

Q9. Find the area of a rhombus whose diagonals are of lengths 9 cm and 7.2 cm.

Solution: Now, d1=9 cm, d2=7.2 cm where d1 and d2 are the lengths of diagonals of a rhombus.

Therefore, area of the given rhombus = ½ x 9 x 7.2 cm² = 32.4 cm²

Note: Here, the value of π is either considered as 22/7 or 3.14, r means radius, and h means height.

Q10. A rectangular paper of a width of 10 cm is rolled along its width and a cylinder of a radius of 18 cm is formed. Find the volume of the cylinder. (Take π=22/7)

Solution: A cylinder is formed by rolling a rectangle of 10 cm about its width. The radius of the cylinder is 18 cm and the width of the paper becomes the height.

Height of the cylinder, h = 10 cm and Radius of the cylinder, r = 18 cm

Now, volume of the cylinder, V = πr²h = (22/7) x (18)² x 10

= 10,183 cm³

Hence, the volume of the cylinder is 10183 cm³. (Mensuration Formula PDF Download)

### Also Check:-

Q.1: What is the perimeter (in cm) of an equilateral triangle whose height is 3.46 cm ?Take \sqrt 3= 1.73.

(A) 12
(B) 9
(C) 6
(D) 10.4

Ans : (A) 12
height of equilateral \triangle =\frac {\sqrt 3}{2} \times side
=\frac {\sqrt 3}{2} \times a = 3.46
\frac {1.73}{2}\times a = 3.46
a = 2 x 2 = 4
Peimeter = 3a = 4 x 3 = 12

Q.2: The perimeters of two similar triangles ABC and PQR are 156 cm and 46.8 cm respectively. If BC = 19.5 cm and QR = x cm, then the value of x is:

(A) 3.76 cm
(B) 5.85 cm
(C) 4.29 cm
(D) 6.75 cm

Ans : (B) 5.85 cm
\frac {(\triangle \text{ABC}) Perimeter}{(\triangle \text{PQR})Perimeter}= \frac {BC}{QR}
\frac {156}{46.8}=\frac {19.5}{x}
x=\frac{46.8\times 19.5}{156}
5.85 cm

Q.3: The radius of a sphere is 9 cm. It is melted and drawn into a wire of radius 0.3 cm. The length of the wire is:

(A) 112 m
(B) 108 m
(C) 118 m
(D) 106 m

Ans : (B) 108 m
Volume of sphere = Wire volume
\frac {4}{3} \pi r^3 = \pir2h
\frac 43 \times \pi \times 9^3 = \pix 0.3 x 0.3 x h
h =\frac {4\times9\times9\times9}{3\times0.3\times0.3}

Q.4: One side of a rectangular field is 39 m and its diagonal is 89 m. What is the area of the field ?

(A) 3120 m2
(B) 2100 m2
(C) 2160 m2
(D) 3140 m2

Ans : (A) 3120 m2
Side = 39
Diagonal = 89
Second side = \sqrt{(89)^2 – (39)^2}
= \sqrt{7921-1521}
Second Side = \sqrt{6400} =80
Area of field = First Side x Second side (l x b)

Q.5: The base of a triangle to the perimeter fo a square whose diagonal is 7\sqrt2 cm and its height is equal to the side of square whose area is 169 cm2 . The area (in cm2 ) of the triangle is:

(A) 130
(B) 182
(C) 175
(D) 156

Ans : (B) 182
Diagonal of square 7\sqrt2
a\sqrt2=7\sqrt2
a = 7
Perimeter of Square = 4a = 28
Perimeter of square = Base of Triangle = 28
Area of Square = 169
Side of Square = 13
Height of Triangle = Side of square = 13
Area of Triangle = \frac12 x b x h
= \frac12 \times28\times13 =182

Q.6: If the adjacent sides of a rectangle whose perimeter is 60 cm are in the ratio 3 : 2, then what will be the area of the rectangle ?

(A) 864 cm2
(B) 216 cm2
(C) 60 cm2
(D) 300 cm2

Ans : (B) 216 cm2
Ration of side = 3 : 2
Perimeter = 60
Perimeter of rectangle = 2 ( l + b)
2(3n + 2n) = 60
5n = 30, n = 6
l = 3n = 3 x 6 = 18
b = 2n = 2 x 6 = 12
Area = l x b = 18 x 12 = 216 cm2 (Mensuration Formula PDF Download)

Q.7: The perimeter of a right angle triangle whose sides that make right angles are 15 cm and 20 cm is:

(A) 60 cm
(B) 40 cm
(C) 70 cm
(D) 50 cm

Ans : (A) 60 cm
Diagonal of Right angle triangle
(Third Side ) =\sqrt{(20^2 + 15^2)}
=\sqrt {400 + 225}
Diagonal =\sqrt {625} = 25
Perimeter = Sum of all sides = 15 + 20 + 25= 60 cm (Mensuration Formula PDF Download)

Q.8: The difference between the two perpendicular sides of a right-angled triangle is 17 cm and its area is 84 cm2 . What is the perimeter (in cm ) of the triangle ?

(A) 65
(B) 49
(C) 72
(D) 56

Ans : (D) 56
b – h = 17
Area = \frac12x b x h = 84
b x h = 168 = 24 x 7
b = 24
h = 7
Diagonal2 = b2 + h2 = 242 + 72
=> 576 + 49 = 625
Diagonal= 25
Sides of Triangle= 7, 24, 25
Perimeter = 7 + 24 + 25 = 56 (Mensuration Formula PDF Download)

Q.9: A solid metallic sphere of radius 10 cm is melted and recast into spheres of radius 2 cm each. How many such spheres can be made ?

(A) 64
(B) 216
(C) 125
(D) 100

Ans : (C) 125
Volume of big sphere = n x Volume of small sphere
V =\frac 43\pi r^3
\frac 43 \pi (10)^3 = n x\frac 43\pi (2)^3
10 x 10 x 10 = n x 2 x 2x 2
1000 = n x 8

Q.10: The sum of the squares of the sides of a rhombus is 1600 cm2 . What is the side of the rhombus ?

(A) 10 cm
(B) 15 cm
(C) 20 cm
(D) 25 cm

Ans : (C) 20 cm
All the sides of rhombus are equal
The sum of the squares of sides = 1600
a2 + a2 + a2 + a2 = 1600
4a2 = 1600
a2 = 400

Q.11: The volume of a right circular cone is 462 cm3 . If its height is 12 cm, then the area of its base (in cm2 ) is:

(A) 124.5
(B) 103.5
(C) 115.5
(D) 98.5

Ans : (C) 115.5
Volume of Cone =\frac13 \pi r^2h
\frac13 \pi r^2h = 462
r2 =\frac{462\times 3}{\pi \times h}
=\frac{462\times 3\times7}{22 \times 12}
r2 =\frac{21\times7}{4}
Area of base =\pi r^2
=\frac {22\times21\times7}{7\times4}

Q.12: How many bricks each measuring 64 cm x 11.25 cm x 6 cm. will be needed to build a wall measuring 8m x 3m x 22.5m ?

(A) 200000
(B) 250000
(C) 67500
(D) 125000

Ans : (D) 125000
Volume of wall = V1
Volume of Brick= V2
Number of Bricks =\frac{V1}{V2}
=\frac{8m\times3m\times22.5}{64cm\times11.25cm\times6cm}
=\frac{800\times300\times22.5}{64\times11.25\times6}

Q.13: If the radius of a circle is equal to a diagonal of a square whose area is 12 cm2 , then the area of the circle is:

(A) 28\pi cm2
(B) 36\pi cm2
(C) 32\pi cm2
(D) 24\pi cm2

Ans : (D) 24\pi cm2
Area of Square = 12
Side of Square =2\sqrt3
Diagonal of square= side x\sqrt2
=2\sqrt6
Radius of Circle = Diagonal of Square
= 2\sqrt6
Area of circle =\pi r^2
=24\pi

Q.14: The sum of three sides of an isosceles triangle is 20 cm, and the ratio of an equal side to the base is 3 : 4, The altitude of the triangle is:

(A) 3\sqrt3cm
(B) 4\sqrt5cm
(C) 3\sqrt5cm
(D) 2\sqrt5cm

Ans : (D) 2\sqrt5cm
Ration of side and base = 3 : 4
Sides 3x, 3x, 4x
Sum of sides = 20
3x + 3x + 4x = 20
10x = 20
x = 2
Sides = 6, 6, 8
Height =\sqrt{6^2 – 4^2}
= 36 – 16
=\sqrt{20}

Q.15: If the volume of a sphere is 697\frac{4}{21}cm3 , then its radius is:
(\text{Take} \pi=\frac{22}{7})

(A) 5 cm
(B) 6 cm
(C) 4.5 cm
(D) 5.5 cm

Ans : (D) 5.5 cm
Volume of Sphere =\frac43\pi r^3
\frac43\pi r^3 = 697\frac{4}{21}
r3 =\frac{14641\times3\times7}{21\times4\times22}
r3 =\frac{1331}{8}
r =\frac{11}{2}

Q.16: The area of the largest that can be inscribed in a semi-circle of radius 6 cm is:

(A) 38cm2
(B) 35cm2
(C) 36cm2
(D) 34cm2

Ans : (C) 36cm2
Base of Tringle = Diameter of semi-circle
Base of Triangle = 12
Height of Triangle = radius of semi-circle
Height= 6
Area of Triangle=\frac12 x base x height
=\frac12 x 12 x 6 = 36 cm2 (Mensuration Formula PDF Download)

Q.17: A solid metallic sphere of radius 12cm is melted and recast in the form of small spheres of radius 2 cm. How many small spheres are formed ?

(A) 96
(B) 864
(C) 216
(D) 24

Ans : (C) 216
Volume of Big sphere= V1
Volume of small sphere= V2
Number of Spheres =\frac{V1}{V2}
=\frac {\frac{4}{3}\pi r^3}{\frac{4}{3}\pi r^3}
=\frac{12\times12\times12}{2\times2\times2}
= 6 x 6 x 6 = 216

Q.18: The square of the diagonal of a cube is 2175 cm2 . What is the total surface area (in cm2 ) of the cube ?

(A) 4272
(B) 4305
(C) 4350
(D) 4530

Ans : (C) 4350
Diagonal of square =a\sqrt3
(a\sqrt3)^2 = 2175
a2 =\frac{2175}{3}
a2 = 725
total surface area = 6a2
= 6 x 725 = 4350

Q.19: A rectangle with perimeter 50 cm has its sides in the ratio 1 : 4. What is the perimeter of a square whose area is the same as that of the rectangle ?

(A) 45 cm
(B) 36 cm
(C) 50 cm
(D) 40 cm

Ans : (D) 40 cm
Let sides are x , 4x
Perimeter = 2(l + b)
2(x + 4x) = 50
5x = 25, x = 5
Length= 20 cm
Breath= 5 cm
Area= l x b = 20 x 5 = 100
As per question, Area of Rectangle = Area of Square
a2 = 100, a = 10
Perimeter of square= 4 x side = 4 x 10 = 40

Q.20: The inner and outer radius of two concentric are 6.7 cm and 9.5 cm, respectively. What is the difference between their circumferences (in cm ) ?\text {Take} \pi=\frac{22}{7}

(A) 10.4
(B) 17.6
(C) 6.5
(D) 20.5

Ans : (B) 17.6
Circumference of circle =2\pir
Difference of Circumference
=2\piR – 2\pir
=2\times\frac{22}{7} [ 9.5 – 6.7]
=2\times\frac{22}{7}\times 2.8
= 17.6