Mensuration Practice Questions & Solved Problems

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16. Mensuration Practice Questions & Solved Problems

In the previous part, we studied the important formulas for Volume and Surface Area of different three-dimensional solids.

This part focuses on applying those formulas to competitive-exam style problems.

The main solids covered are:

  • Cube
  • Cuboid
  • Cylinder
  • Cone
  • Frustum
  • Sphere
  • Hollow Sphere
  • Hemisphere
  • Prism
  • Pyramid

1. How to Solve Mensuration Problems

  1. Identify the solid.
  2. Write the required formula.
  3. Write the given values.
  4. Substitute the values.
  5. Simplify carefully.
  6. Write the correct unit.

2. Cube — Solved Problems

a a a

Question 1

A cube has an edge of 8 cm. Find its volume.

Formula:

V = a³

Given: a = 8 cm

V = 8³
V = 512 cm³

Answer: 512 cm³

Question 2

Find the total surface area of a cube whose edge is 6 cm.

TSA = 6a²
= 6 × 6²
= 6 × 36
TSA = 216 cm²

3. Cuboid — Solved Problems

L H B

Question 3

A cuboid has length 12 cm, breadth 5 cm and height 4 cm. Find its volume.

V = L × B × H
V = 12 × 5 × 4
V = 240 cm³

Question 4

A cuboid has L = 10 cm, B = 6 cm and H = 5 cm. Find its total surface area.

TSA = 2(LB + BH + HL)
= 2(10×6 + 6×5 + 5×10)
= 2(60 + 30 + 50)
= 2 × 140
TSA = 280 cm²

4. Cylinder — Solved Problems

r h

Question 5

A cylinder has radius 7 cm and height 10 cm. Find its volume. Take π = 22/7.

V = πr²h
= 22/7 × 7² × 10
= 22/7 × 49 × 10
V = 1540 cm³

Question 6

Find the curved surface area of a cylinder with radius 5 cm and height 14 cm. Take π = 22/7.

CSA = 2πrh
= 2 × 22/7 × 5 × 14
CSA = 440 cm²

5. Cone — Solved Problems

h r

Question 7

A cone has radius 3 cm and height 4 cm. Find its volume.

V = ⅓πr²h
= ⅓ × π × 3² × 4
= 12π
V = 12π cm³

Question 8

A cone has radius 7 cm and slant height 10 cm. Find its curved surface area. Take π = 22/7.

CSA = πrl
= 22/7 × 7 × 10
CSA = 220 cm²

6. Sphere — Solved Problems

r

Question 9

Find the volume of a sphere of radius 6 cm.

V = ⁴⁄₃πr³
= ⁴⁄₃ × π × 6³
= ⁴⁄₃ × π × 216
V = 288π cm³

Question 10

Find the surface area of a sphere whose radius is 7 cm. Take π = 22/7.

Surface Area = 4πr²
= 4 × 22/7 × 7²
= 4 × 22 × 7
Surface Area = 616 cm²

7. Hemisphere — Solved Problems

r

Question 11

Find the volume of a hemisphere whose radius is 6 cm.

V = ⅔πr³
= ⅔ × π × 6³
= ⅔ × π × 216
V = 144π cm³

Question 12

Find the total surface area of a hemisphere of radius 7 cm. Take π = 22/7.

TSA = 3πr²
= 3 × 22/7 × 49
TSA = 462 cm²

8. Hollow Sphere — Solved Problem

R r

Question 13

A hollow sphere has outer radius 5 cm and inner radius 3 cm. Find its volume.

V = ⁴⁄₃π(R³ − r³)
= ⁴⁄₃π(5³ − 3³)
= ⁴⁄₃π(125 − 27)
= ⁴⁄₃π × 98
V = 392π/3 cm³

9. Frustum — Solved Problem

Question 14

A frustum has larger radius R = 5 cm, smaller radius r = 3 cm, and height h = 4 cm. Find its volume.

V = ⅓π(R² + r² + Rr)h
= ⅓π(25 + 9 + 15) × 4
= ⅓π × 49 × 4
V = 196π/3 cm³

10. Prism — Solved Problem

Question 15

A prism has a base area of 25 cm² and height 12 cm. Find its volume.

V = Base Area × Height
= 25 × 12
V = 300 cm³

11. Pyramid — Solved Problem

Question 16

A pyramid has a base area of 45 cm² and height 12 cm. Find its volume.

V = ⅓ × Base Area × Height
= ⅓ × 45 × 12
= 15 × 12
V = 180 cm³

12. Mixed Mensuration Practice Set

Try to solve the following questions before checking the answers.

1. Find the volume of a cube of side 9 cm.

2. Find the TSA of a cube of side 10 cm.

3. A cuboid has dimensions 15 cm × 8 cm × 4 cm. Find its volume.

4. Find the TSA of a cuboid with L = 10 cm, B = 5 cm and H = 3 cm.

5. Find the volume of a cylinder with r = 7 cm and h = 8 cm. Take π = 22/7.

6. Find the CSA of a cylinder with r = 7 cm and h = 12 cm. Take π = 22/7.

7. Find the volume of a cone with r = 6 cm and h = 9 cm.

8. Find the volume of a sphere with radius 3 cm.

9. Find the volume of a hemisphere with radius 7 cm. Take π = 22/7.

10. A prism has base area 35 cm² and height 8 cm. Find its volume.

13. Answers — Quick Check

Question Answer
1 729 cm³
2 600 cm²
3 480 cm³
4 190 cm²
5 1232 cm³
6 528 cm²
7 108π cm³
8 36π cm³
9 2156/3 cm³
10 280 cm³

14. Important Formula Shortcuts

Cube → V = a³
Cuboid → V = LBH
Cylinder → V = πr²h
Cone → V = ⅓πr²h
Sphere → V = ⁴⁄₃πr³
Hemisphere → V = ⅔πr³
Frustum → V = ⅓π(R²+r²+Rr)h
Prism → V = Base Area × Height
Pyramid → V = ⅓ × Base Area × Height

15. Common Exam Mistakes

1. Radius vs Diameter

If diameter is given, first convert it into radius:

r = d/2

2. Area vs Volume

Surface area is expressed in: cm², m², etc.

Volume is expressed in: cm³, m³, etc.

3. Cone and Pyramid

Do not forget the factor 1/3.

4. Cylinder

Remember that:

CSA = 2πrh

while:

TSA = 2πr(r+h)

5. Hemisphere

For total surface area, the circular base must also be included:

TSA = 3πr²

16. SSC CGL & Government Exam Strategy

For competitive examinations, first memorise the basic volume formulas:

  1. Cube
  2. Cuboid
  3. Cylinder
  4. Cone
  5. Sphere
  6. Hemisphere
  7. Frustum
  8. Prism
  9. Pyramid

Then practise questions involving:

  • Volume
  • CSA
  • LSA
  • TSA
  • Radius and diameter conversion
  • Height and slant height
  • Unit conversion
  • Mixed solids

17. Final Revision

Step 1: Identify the solid.

Step 2: Write the formula.

Step 3: Substitute the values.

Step 4: Calculate carefully.

Step 5: Check the unit.

Area → square units
Volume → cubic units

Source note: The formulas used in this practice section are based on the Volume & Surface Area formulas in the provided PDF. The PDF itself ends with Cube, Cuboid, Cylinder, Cone, Frustum, Sphere, Hollow Sphere, Hemisphere, Prism and Pyramid formulas. :contentReference[oaicite:1]{index=1}

Part 16 status: This is a practice extension created from those PDF formulas; the practice questions themselves are not stated in the PDF.