Quick Summary
Mensuration covers the measurement of 2D shapes (perimeter and area) and 3D solids (surface area and volume). This chapter requires careful formula memorisation and unit conversion. Problems involving combinations of shapes (e.g., cylinder surmounted by a cone) are common in 5-mark questions.
2D Figures — Formulae
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | (1/2) × b × h | a + b + c |
| Equilateral Triangle | (√3/4)a² | 3a |
| Circle | πr² | 2πr (circumference) |
| Semi-circle | (1/2)πr² | πr + 2r |
| Parallelogram | b × h | 2(a + b) |
| Trapezium | (1/2)(a + b) × h | sum of all sides |
Heron's Formula (when height is not given): Area = √(s(s−a)(s−b)(s−c)) where s = (a + b + c)/2
3D Solids — Surface Area Formulae
| Solid | LSA | TSA |
|---|---|---|
| Cuboid | 2h(l + b) | 2(lb + bh + lh) |
| Cube | 4a² | 6a² |
| Cylinder | 2πrh | 2πr(r + h) |
| Cone | πrl | πr(r + l) where l = slant height |
| Sphere | — | 4πr² |
| Hemisphere | 2πr² | 3πr² |
Slant height of cone: l = √(r² + h²)
3D Solids — Volume Formulae
| Solid | Volume |
|---|---|
| Cuboid | l × b × h |
| Cube | a³ |
| Cylinder | πr²h |
| Cone | (1/3)πr²h |
| Sphere | (4/3)πr³ |
| Hemisphere | (2/3)πr³ |
Important Questions
One Mark
1. Find the area of a circle with radius 7 cm. (Use π = 22/7) 2. What is the slant height of a cone with r = 3 cm and h = 4 cm? 3. Find the volume of a cube with side 5 cm. 4. State Heron's formula. 5. Find the TSA of a sphere with radius 6 cm.
Short Answer — 3 marks
1. Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm using Heron's formula. 2. The radius of a cylinder is 7 cm and height is 10 cm. Find its LSA and volume. 3. A cone has a base radius of 5 cm and slant height of 13 cm. Find its TSA and volume.
Long Answer — 5 marks
1. A toy is made by placing a cone on top of a cylinder. The cylinder has radius 3 cm and height 10 cm. The cone has the same base radius and height 4 cm. Find the total surface area and volume of the toy. 2. A rectangular park 60 m × 40 m has a circular pond of radius 7 m in its centre. Find the area of the park excluding the pond. Also find the cost of fencing the park at ₹12 per metre.
Common Mistakes
- Using diameter instead of radius in circle/sphere/cylinder formulae.
- Forgetting the factor (1/3) in cone volume.
- Confusing LSA and TSA — not adding the base circles for TSA of cylinder/cone.
- Not converting units before calculation (e.g., cm to m).
Revision Checklist
- [ ] Memorised all 2D area and perimeter formulae
- [ ] Know Heron's formula and can apply it
- [ ] Memorised LSA, TSA, and Volume for all five 3D solids
- [ ] Can find slant height of a cone using Pythagoras
- [ ] Practised at least one combination-shape problem
- [ ] Confident with unit conversion (cm²/m², cm³/m³)