Formulas to know
| Shape | Area or surface area | Perimeter or volume |
|---|---|---|
| Rectangle (l, b) | l × b | 2(l + b) |
| Square (a) | a² | 4a |
| Triangle (base b, height h) | ½ b h | sum of sides |
| Circle (r) | πr² | 2πr |
| Cuboid (l, b, h) | 2(lb + bh + hl) | l b h |
| Cube (a) | 6a² | a³ |
| Cylinder (r, h) | 2πr(r + h) | πr²h |
| Cone (r, h, slant l) | πr(r + l) | ⅓πr²h |
| Sphere (r) | 4πr² | (4/3)πr³ |
Use π = 22/7 unless the question says otherwise; check the instruction for each exam.
Worked examples
Example 1. The sides of a rectangle are in the ratio 3 : 2 and its perimeter is 50 cm. The sum of adjacent sides is 25, so the sides are 15 and 10; area = 150 cm².
Example 2. A circle has circumference 44 cm. 2πr = 44 with π = 22/7 gives r = 7 cm; area = (22/7) × 49 = 154 cm².
Example 3 (cylinder). A cylinder has radius 7 cm and height 10 cm. Volume = (22/7) × 49 × 10 = 1,540 cm³. Curved surface area = 2πrh = 2 × (22/7) × 7 × 10 = 440 cm².
Example 4 (changing dimensions). If the radius of a circle increases by 10%, its area increases by 1.1² − 1 = 21%.
Example 5 (melting and recasting). A solid metal sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. Volumes ∝ r³: (6 ÷ 2)³ = 27 small spheres.
Example 6 (triangle). A right triangle has legs 6 and 8, so the hypotenuse is 10 and the area is ½ × 6 × 8 = 24.
Example 7 (cube). A cube has a surface area of 150 cm². 6a² = 150 gives a² = 25, a = 5; volume = 125 cm³.
Common traps
- Mixing units: convert all lengths to the same unit before computing.
- Using diameter for radius.
- Confusing curved and total surface area.
- Forgetting that area scales with the square and volume with the cube of linear dimensions.
Speed tips
- Memorise Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
- Use ratios of similar shapes: area ratio = (side ratio)², volume ratio = (side ratio)³.
- Keep π as 22/7 and cancel with 7s.
A practice routine
- Day 1: 2D area and perimeter, ten questions.
- Day 2: circles and sectors.
- Day 3: 3D shapes and surface area.
- Day 4: melting and recasting, and a timed set of 12 at 60 seconds each.
Where it appears
Mensuration appears in SSC CGL, railway mathematics, MAT and CLAT papers. For speed with numbers, revise squares, cubes and roots.
Practice tests below are published by institutes and educators on TestMint.
