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Mensuration

Area, perimeter, surface area and volume.

45 solved questions · 15 easy · 20 medium · 10 hard

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Shortcuts & formulas

  • Rectangle: area = l x b, perimeter = 2(l + b). Triangle: area = 1/2 x base x height; Heron: sqrt(s(s-a)(s-b)(s-c)).
  • Circle: area = pi r^2, circumference = 2 pi r. Use pi = 22/7 when the radius is a multiple of 7.
  • Equilateral triangle: area = (sqrt3/4) a^2. Rhombus: area = 1/2 x d1 x d2. Trapezium: 1/2 (a + b) h.
  • Cuboid: V = lbh, TSA = 2(lb + bh + hl), diagonal = sqrt(l^2 + b^2 + h^2). Cube: V = a^3, TSA = 6a^2, diagonal = a sqrt3.
  • Cylinder: V = pi r^2 h, CSA = 2 pi r h, TSA = 2 pi r (r + h).
  • Cone: V = 1/3 pi r^2 h, slant l = sqrt(r^2 + h^2), CSA = pi r l, TSA = pi r (r + l).
  • Sphere: V = 4/3 pi r^3, S = 4 pi r^2. Hemisphere: V = 2/3 pi r^3, CSA = 2 pi r^2, TSA = 3 pi r^2.
  • When a solid is melted and recast, volume stays the same. 1 m^3 = 1000 litres.

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

Find the area of a rectangle of length 25 cm and breadth 12 cm.

  1. A300 cm^2
  2. B74 cm^2
  3. C37 cm^2
  4. D150 cm^2
Show answer & solution

Answer: A. 300 cm^2

Solution

Area = l x b = 25 x 12 = 300 cm^2 Answer: 300 cm^2

Q2.

The perimeter of a rectangular field is 96 m and its length is 30 m. Find its breadth.

  1. A66 m
  2. B18 m
  3. C36 m
  4. D48 m
Show answer & solution

Answer: B. 18 m

Solution

2(l + b) = 96, so l + b = 48 b = 48 - 30 = 18 m Answer: 18 m

Q3.

Find the area of a circle of radius 14 cm. (Use pi = 22/7)

  1. A88 cm^2
  2. B1,232 cm^2
  3. C616 cm^2
  4. D308 cm^2
Show answer & solution

Answer: C. 616 cm^2

Solution

Area = pi r^2 = (22/7) x 14 x 14 = 22 x 2 x 14 = 616 cm^2 Answer: 616 cm^2

Q4.

Find the circumference of a circle of diameter 35 cm. (Use pi = 22/7)

  1. A220 cm
  2. B963 cm
  3. C55 cm
  4. D110 cm
Show answer & solution

Answer: D. 110 cm

Solution

Circumference = pi d = (22/7) x 35 = 110 cm Answer: 110 cm

Q5.

Find the volume of a cuboid of length 12 cm, breadth 5 cm and height 4 cm.

  1. A240 cm^3
  2. B256 cm^3
  3. C21 cm^3
  4. D128 cm^3
Show answer & solution

Answer: A. 240 cm^3

Solution

Volume = l x b x h = 12 x 5 x 4 = 240 cm^3 Answer: 240 cm^3

Q6.

Find the total surface area of a cube of side 9 cm.

  1. A324 cm^2
  2. B486 cm^2
  3. C729 cm^2
  4. D243 cm^2
Show answer & solution

Answer: B. 486 cm^2

Solution

TSA = 6a^2 = 6 x 81 = 486 cm^2 Answer: 486 cm^2

Q7.

Find the volume of a cylinder of radius 7 cm and height 10 cm. (Use pi = 22/7)

  1. A440 cm^3
  2. B4,620 cm^3
  3. C1,540 cm^3
  4. D770 cm^3
Show answer & solution

Answer: C. 1,540 cm^3

Solution

Volume = pi r^2 h = (22/7) x 49 x 10 = 22 x 7 x 10 = 1,540 cm^3 Answer: 1,540 cm^3

Q8.

Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.

  1. A91 cm^2
  2. B105 cm^2
  3. C42 cm^2
  4. D84 cm^2
Show answer & solution

Answer: D. 84 cm^2

Solution

s = (13 + 14 + 15)/2 = 21 Area = sqrt(21 x 8 x 7 x 6) = sqrt(7056) = 84 cm^2 Answer: 84 cm^2

Shortcut: Learn the 13-14-15 triangle: area 84.

Q9.

The parallel sides of a trapezium are 12 cm and 18 cm and the distance between them is 8 cm. Find its area.

  1. A120 cm^2
  2. B240 cm^2
  3. C144 cm^2
  4. D96 cm^2
Show answer & solution

Answer: A. 120 cm^2

Solution

Area = 1/2 x (a + b) x h = 1/2 x 30 x 8 = 120 cm^2 Answer: 120 cm^2

Q10.

The diagonals of a rhombus are 16 cm and 30 cm. Find its area.

  1. A480 cm^2
  2. B240 cm^2
  3. C46 cm^2
  4. D120 cm^2
Show answer & solution

Answer: B. 240 cm^2

Solution

Area = 1/2 x d1 x d2 = 1/2 x 16 x 30 = 240 cm^2 Answer: 240 cm^2

Q11.

The volume of a cube is 1728 cm^3. Find the length of its edge.

  1. A24 cm
  2. B42 cm
  3. C12 cm
  4. D9 cm
Show answer & solution

Answer: C. 12 cm

Solution

a^3 = 1728 = 12^3 So a = 12 cm Answer: 12 cm

Q12.

Find the curved surface area of a cylinder of radius 7 cm and height 20 cm. (Use pi = 22/7)

  1. A1,760 cm^2
  2. B440 cm^2
  3. C3,080 cm^2
  4. D880 cm^2
Show answer & solution

Answer: D. 880 cm^2

Solution

CSA = 2 pi r h = 2 x (22/7) x 7 x 20 = 880 cm^2 Answer: 880 cm^2

Q13.

Find the volume of a sphere of radius 21 cm. (Use pi = 22/7)

  1. A38,808 cm^3
  2. B5,544 cm^3
  3. C29,106 cm^3
  4. D19,404 cm^3
Show answer & solution

Answer: A. 38,808 cm^3

Solution

V = 4/3 pi r^3 = (4/3) x (22/7) x 21 x 21 x 21 = 4 x 22 x 7 x 21 x 3 / 3... compute: 21^3 = 9261; (4/3) x 9261 = 12348; 12348 x 22/7 = 38,808 cm^3 Answer: 38,808 cm^3

Q14.

Find the volume of a cone of base radius 7 cm and height 15 cm. (Use pi = 22/7)

  1. A2,310 cm^3
  2. B770 cm^3
  3. C1,540 cm^3
  4. D385 cm^3
Show answer & solution

Answer: B. 770 cm^3

Solution

V = 1/3 pi r^2 h = (1/3) x (22/7) x 49 x 15 = 22 x 7 x 5 = 770 cm^3 Answer: 770 cm^3

Q15.

Find the surface area of a sphere of radius 7 cm. (Use pi = 22/7)

  1. A154 cm^2
  2. B308 cm^2
  3. C616 cm^2
  4. D1,437 cm^2
Show answer & solution

Answer: C. 616 cm^2

Solution

S = 4 pi r^2 = 4 x (22/7) x 49 = 616 cm^2 Answer: 616 cm^2

Medium questions

Q16.

A rectangular garden 40 m long and 30 m wide has a 2 m wide path running all around it on the outside. Find the area of the path.

  1. A280 m^2
  2. B336 m^2
  3. C1,496 m^2
  4. D296 m^2
Show answer & solution

Answer: D. 296 m^2

Solution

Outer rectangle = (40 + 4) x (30 + 4) = 44 x 34 = 1,496 m^2 Garden = 40 x 30 = 1,200 m^2 Path = 1,496 - 1,200 = 296 m^2 Answer: 296 m^2

Q17.

The perimeter of a rectangular plot 30 m by 20 m is to be fenced at a rate of ₹12 per metre. Find the total cost of fencing.

  1. A₹1,200
  2. B₹7,200
  3. C₹600
  4. D₹2,400
Show answer & solution

Answer: A. ₹1,200

Solution

Perimeter = 2(30 + 20) = 100 m Cost = 100 x 12 = ₹1,200 Answer: ₹1,200

Q18.

A room is 10 m long, 8 m wide and 4 m high. It has one door of 2 m x 1 m and one window of 1 m x 1 m. Find the area of the four walls to be painted, excluding the door and window.

  1. A144 m^2
  2. B138 m^2
  3. C142 m^2
  4. D221 m^2
Show answer & solution

Answer: B. 138 m^2

Solution

Area of 4 walls = 2(l + b) x h = 2 x 18 x 4 = 144 m^2 Door + window = 2 + 1 = 3 m^2 Paintable area = 144 - 3 = 141 m^2 Answer: 141 m^2

Q19.

A wheel has a diameter of 70 cm. How many complete revolutions does it make to cover 1.1 km? (Use pi = 22/7)

  1. A250
  2. B1000
  3. C500
  4. D5000
Show answer & solution

Answer: C. 500

Solution

Circumference = pi d = (22/7) x 70 = 220 cm Distance = 1.1 km = 110,000 cm Revolutions = 110000/220 = 500 Answer: 500

Q20.

Find the length of the diagonal of a cuboid whose dimensions are 3 cm, 4 cm and 12 cm.

  1. A19 cm
  2. B12 cm
  3. C17 cm
  4. D13 cm
Show answer & solution

Answer: D. 13 cm

Solution

Diagonal = sqrt(l^2 + b^2 + h^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13 cm Answer: 13 cm

Q21.

A solid metal sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. How many small spheres are formed?

  1. A27
  2. B9
  3. C3
  4. D18
Show answer & solution

Answer: A. 27

Solution

Volume is conserved. Number = (6^3)/(2^3) = 216/8 = 27 Answer: 27

Shortcut: Number = (R/r)^3 = 3^3 = 27

Q22.

The radius of a circle is increased by 10%. By what percentage does its area increase?

  1. A10%
  2. B21%
  3. C20%
  4. D11%
Show answer & solution

Answer: B. 21%

Solution

New area = pi (1.1 r)^2 = 1.21 pi r^2 Increase = 21% Answer: 21%

Shortcut: x + x + x^2/100 = 10 + 10 + 1 = 21

Q23.

Find the total surface area of a solid hemisphere of radius 7 cm. (Use pi = 22/7)

  1. A308 cm^2
  2. B616 cm^2
  3. C462 cm^2
  4. D154 cm^2
Show answer & solution

Answer: C. 462 cm^2

Solution

TSA = 3 pi r^2 = 3 x (22/7) x 49 = 462 cm^2 (Curved part 2 pi r^2 = 308, plus flat base pi r^2 = 154) Answer: 462 cm^2

Q24.

Find the area of a sector of a circle of radius 14 cm and central angle 90 degrees. (Use pi = 22/7)

  1. A616 cm^2
  2. B308 cm^2
  3. C77 cm^2
  4. D154 cm^2
Show answer & solution

Answer: D. 154 cm^2

Solution

Area = (theta/360) x pi r^2 = (90/360) x (22/7) x 196 = 616/4 = 154 cm^2 Answer: 154 cm^2

Q25.

Find the area of an equilateral triangle of side 12 cm.

  1. A36 sqrt3 cm^2
  2. B72 sqrt3 cm^2
  3. C144 sqrt3 cm^2
  4. D18 sqrt3 cm^2
Show answer & solution

Answer: A. 36 sqrt3 cm^2

Solution

Area = (sqrt3/4) a^2 = (sqrt3/4) x 144 = 36 sqrt3 cm^2 Answer: 36 sqrt3 cm^2

Q26.

A rectangular tank is 3 m long, 2 m wide and 1.5 m deep. How many litres of water can it hold when full?

  1. A900 litres
  2. B9,000 litres
  3. C90,000 litres
  4. D3,000 litres
Show answer & solution

Answer: B. 9,000 litres

Solution

Volume = 3 x 2 x 1.5 = 9 m^3 1 m^3 = 1000 litres Capacity = 9 x 1000 = 9,000 litres Answer: 9,000 litres

Q27.

A circular ring (annulus) has an outer radius of 21 cm and an inner radius of 14 cm. Find its area. (Use pi = 22/7)

  1. A154 cm^2
  2. B1,386 cm^2
  3. C770 cm^2
  4. D616 cm^2
Show answer & solution

Answer: C. 770 cm^2

Solution

Area = pi (R^2 - r^2) = (22/7)(441 - 196) = (22/7) x 245 = 770 cm^2 Answer: 770 cm^2

Q28.

Find the curved surface area of a cone with base radius 7 cm and height 24 cm. (Use pi = 22/7)

  1. A528 cm^2
  2. B682 cm^2
  3. C3,850 cm^2
  4. D550 cm^2
Show answer & solution

Answer: D. 550 cm^2

Solution

Slant height l = sqrt(7^2 + 24^2) = sqrt(625) = 25 cm CSA = pi r l = (22/7) x 7 x 25 = 550 cm^2 Answer: 550 cm^2

Q29.

The diagonal of a cube is 5 sqrt3 cm. Find its volume.

  1. A125 cm^3
  2. B75 cm^3
  3. C25 cm^3
  4. D375 cm^3
Show answer & solution

Answer: A. 125 cm^3

Solution

Diagonal = a sqrt3, so a = 5 cm Volume = a^3 = 125 cm^3 Answer: 125 cm^3

Q30.

The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Find the ratio of their volumes.

  1. A10 : 9
  2. B20 : 27
  3. C4 : 9
  4. D10 : 27
Show answer & solution

Answer: B. 20 : 27

Solution

Volume ratio = r1^2 h1 : r2^2 h2 = (4 x 5) : (9 x 3) = 20 : 27 Answer: 20 : 27

Q31.

Find the perimeter of a semicircular plate of radius 7 cm (include the straight edge). (Use pi = 22/7)

  1. A22 cm
  2. B44 cm
  3. C36 cm
  4. D58 cm
Show answer & solution

Answer: C. 36 cm

Solution

Curved part = pi r = (22/7) x 7 = 22 cm Straight edge = diameter = 14 cm Perimeter = 22 + 14 = 36 cm Answer: 36 cm

Q32.

A circular ground has a circumference of 88 m. Find the cost of levelling it at ₹5 per square metre. (Use pi = 22/7)

  1. A₹440
  2. B₹220
  3. C₹6,160
  4. D₹3,080
Show answer & solution

Answer: D. ₹3,080

Solution

2 pi r = 88, so r = 88 x 7/(2 x 22) = 14 m Area = (22/7) x 14 x 14 = 616 m^2 Cost = 616 x 5 = ₹3,080 Answer: ₹3,080

Q33.

The length of a rectangle is increased by 20% and its breadth is decreased by 10%. What is the percentage change in its area?

  1. A8% increase
  2. B10% increase
  3. C6% increase
  4. D12% increase
Show answer & solution

Answer: A. 8% increase

Solution

New area = 1.2 x 0.9 = 1.08 of the original Increase = 8% Answer: 8% increase

Shortcut: +20 - 10 - (20 x 10)/100 = +8

Q34.

The side of a cube is doubled. By what percentage does its total surface area increase?

  1. A100%
  2. B200%
  3. C700%
  4. D300%
Show answer & solution

Answer: D. 300%

Solution

Surface area is proportional to a^2. New area = (2a)^2 = 4a^2 = 4 times the original. Increase = (4 - 1) x 100 = 300% Answer: 300%

Q35.

A cylindrical water tank has a radius of 7 m and a height of 10 m. How many litres of water can it hold? (Use pi = 22/7)

  1. A15,40,000 litres
  2. B1,540 litres
  3. C1,54,000 litres
  4. D61,60,000 litres
Show answer & solution

Answer: A. 15,40,000 litres

Solution

Volume = pi r^2 h = (22/7) x 49 x 10 = 1,540 m^3 1 m^3 = 1000 litres Capacity = 1,540 x 1000 = 15,40,000 litres Answer: 15,40,000 litres

Hard questions

Q36.

A cone of the largest possible volume is carved out of a solid cylinder of radius 7 cm and height 12 cm. How much material is wasted? (Use pi = 22/7)

  1. A616 cm^3
  2. B1,232 cm^3
  3. C924 cm^3
  4. D1,848 cm^3
Show answer & solution

Answer: B. 1,232 cm^3

Solution

Cylinder volume = (22/7) x 49 x 12 = 1,848 cm^3 Largest cone has the same base and height: volume = 1/3 of cylinder = 616 cm^3 Wasted = 1848 - 616 = 1,232 cm^3 Answer: 1,232 cm^3

Shortcut: Waste = 2/3 of the cylinder.

Q37.

A solid sphere of radius 9 cm is melted and recast into a cone of base radius 18 cm. Find the height of the cone.

  1. A3 cm
  2. B6 cm
  3. C9 cm
  4. D12 cm
Show answer & solution

Answer: C. 9 cm

Solution

Volume of sphere = (4/3) pi (9^3) = (4/3) x 729 pi = 972 pi Volume of cone = (1/3) pi (18^2) h = 108 pi h 108 h = 972, so h = 9 Answer: 9 cm

Q38.

A sector of a circle of radius 21 cm and central angle 120 degrees is rolled up to form a cone. Find the radius of the base of the cone.

  1. A14 cm
  2. B10.5 cm
  3. C15 cm
  4. D7 cm
Show answer & solution

Answer: D. 7 cm

Solution

Slant height of cone = radius of sector = 21 cm Arc length of sector = (120/360) x 2 pi x 21 = 14 pi This becomes the base circumference: 2 pi r = 14 pi, so r = 7 cm Answer: 7 cm

Shortcut: r = l x (angle/360) = 21 x 1/3 = 7

Q39.

A bucket in the shape of a frustum of a cone is 9 cm high with base radii 14 cm and 7 cm. Find its capacity. (Use pi = 22/7)

  1. A3,234 cm^3
  2. B2,548 cm^3
  3. C5,544 cm^3
  4. D2,156 cm^3
Show answer & solution

Answer: A. 3,234 cm^3

Solution

V = (1/3) pi h (R^2 + r^2 + Rr) = (1/3) x (22/7) x 9 x (196 + 49 + 98) = (22/7) x 3 x 343 = 22 x 3 x 49 = 3,234 cm^3 Answer: 3,234 cm^3

Q40.

Water flows through a circular pipe of diameter 14 cm at a speed of 5 m/s into a rectangular tank 7 m long, 6 m wide and 1.1 m deep. How many minutes will it take to fill the empty tank? (Use pi = 22/7)

  1. A5 minutes
  2. B10 minutes
  3. C20 minutes
  4. D50 minutes
Show answer & solution

Answer: B. 10 minutes

Solution

Radius = 7 cm = 0.07 m Water per second = pi r^2 x speed = (22/7) x 0.0049 x 5 = 0.077 m^3 Tank volume = 7 x 6 x 1.1 = 46.2 m^3 Time = 46.2/0.077 = 600 seconds = 10 minutes Answer: 10 minutes

Q41.

A square is inscribed in a circle of radius 7 cm. Find the area of the circle that lies outside the square. (Use pi = 22/7)

  1. A105 cm^2
  2. B154 cm^2
  3. C56 cm^2
  4. D98 cm^2
Show answer & solution

Answer: C. 56 cm^2

Solution

Diagonal of square = diameter = 14 cm Area of square = d^2/2 = 196/2 = 98 cm^2 Area of circle = (22/7) x 49 = 154 cm^2 Area outside the square = 154 - 98 = 56 cm^2 Answer: 56 cm^2

Q42.

A hollow metal pipe is 14 cm long with outer radius 5 cm and inner radius 4 cm. Find the volume of metal in it. (Use pi = 22/7)

  1. A792 cm^3
  2. B44 cm^3
  3. C1,100 cm^3
  4. D396 cm^3
Show answer & solution

Answer: D. 396 cm^3

Solution

Volume of metal = pi (R^2 - r^2) h = (22/7)(25 - 16) x 14 = 22 x 9 x 2 = 396 cm^3 Answer: 396 cm^3

Q43.

The radius of a cylinder is increased by 20% and its height is decreased by 25%. What is the percentage change in its volume?

  1. A8% increase
  2. B5% increase
  3. C15% increase
  4. D10% increase
Show answer & solution

Answer: A. 8% increase

Solution

New volume = (1.2)^2 x 0.75 = 1.44 x 0.75 = 1.08 of the original Increase = 8% Answer: 8% increase

Q44.

A cone and a hemisphere stand on the same base of radius 7 cm and have equal volumes. Find the height of the cone.

  1. A7 cm
  2. B14 cm
  3. C21 cm
  4. D4.67 cm
Show answer & solution

Answer: B. 14 cm

Solution

Cone volume = (1/3) pi r^2 h; hemisphere volume = (2/3) pi r^3 Equal: (1/3) h = (2/3) r, so h = 2r = 14 cm Answer: 14 cm

Q45.

What is the ratio of the volume of a cube to the volume of the largest sphere that can be placed inside it? (Use pi = 22/7)

  1. A11 : 21
  2. B42 : 11
  3. C21 : 11
  4. D7 : 11
Show answer & solution

Answer: C. 21 : 11

Solution

Let the cube have side 2r; the largest sphere has radius r. Cube volume = 8 r^3; sphere volume = (4/3)(22/7) r^3 = (88/21) r^3 Ratio = 8 : 88/21 = 168 : 88 = 21 : 11 Answer: 21 : 11

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