# Mensuration — Area, Perimeter, Volume and Surface Area (2D and 3D)

Source: https://skills101.in/learn/aptitude/mensuration — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

Mensuration means **measuring shapes**: how much space they cover (area), the length around them (perimeter), and for solid objects how much they hold (volume) and how much wrapping they need (surface area).

## 1. 2D shapes

![Formula card: rectangle l×b, square a², circle πr², triangle ½×b×h, and Pythagoras c² = a² + b²](https://skills101.in/learn-img/aptitude/shapes-2d.svg)

| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Triangle | ½ × base × height | sum of sides |
| Equilateral triangle | (√3/4) a² | 3a |
| Circle | πr² | 2πr (circumference) |
| Semicircle | ½πr² | πr + 2r |
| Parallelogram | base × height | 2(a + b) |
| Trapezium | ½ × (sum of parallel sides) × height | sum of sides |
| Rhombus | ½ × d₁ × d₂ | 4a |

**Diagonal of a square = a√2. Diagonal of a rectangle = √(l² + b²).**

### Example 1
*A rectangular field is 40 m by 30 m. Find its area, perimeter and diagonal.*

Area = **1200 m²**. Perimeter = 2(70) = **140 m**. Diagonal = √(1600 + 900) = √2500 = **50 m**.

### Example 2
*A wire bent into a square encloses 484 cm². The same wire is bent into a circle. Find the circle's area. (π = 22/7)*

Side = 22 → wire = 88 cm = 2πr → r = 88 × 7 ÷ 44 = 14 → area = 22/7 × 14 × 14 = **616 cm²**.

### Example 3 — wheels
*A wheel of radius 35 cm makes how many turns to cover 1.1 km?*

One turn = 2πr = 2 × 22/7 × 35 = 220 cm. 1.1 km = 110,000 cm → turns = **500**.

## 2. 3D shapes

![Formula table for cube, cuboid, cylinder, cone, sphere and hemisphere: volume and total surface area](https://skills101.in/learn-img/aptitude/shapes-3d.svg)

Extra formulas:
- Cuboid diagonal = √(l² + b² + h²). Cube diagonal = a√3.
- Cylinder **curved** surface area = 2πrh. Cone curved surface = πrl, where **slant height l = √(r² + h²)**.

### Example 4
*Find the volume of a cylindrical tank of radius 7 m and height 10 m, in litres.*

V = 22/7 × 49 × 10 = 1540 m³. 1 m³ = 1000 litres → **15,40,000 litres**.

### Example 5 — melting and recasting
*A metal sphere of radius 6 cm is melted into small spheres of radius 2 cm. How many small spheres?*

Volume is the same: (6/2)³ = **27** small spheres. (Volume ∝ r³, so you do not need π.)

### Example 6
*How many 10 cm cubes fit in a box 1 m × 50 cm × 40 cm?*

(100 × 50 × 40) ÷ (10 × 10 × 10) = 200,000 ÷ 1000 = **200**.

## 3. Percentage change in dimensions

- If every **length** grows by x%, **area** grows by 2x + x²/100 % (successive change twice).
- **Volume** grows by about three times as much (apply the change three times).

### Example 7
*The radius of a circle increases by 10%. By what % does the area increase?*

10 + 10 + 1 = **21%**.

### Example 8
*A rectangle's length increases by 20% and breadth decreases by 10%. Change in area?*

20 − 10 + (20 × −10)/100 = 10 − 2 = **+8%**.

## Common mistakes

- Mixing units: cm and m in the same formula. Convert first (1 m = 100 cm, 1 m² = 10,000 cm², 1 m³ = 1000 L).
- Using height instead of slant height in the cone's surface area.
- Thinking area doubles when sides double. It becomes **4 times** (and volume 8 times).

## Quick recap

- Learn the two picture cards.
- Recasting → volumes are equal; volume ∝ r³.
- Wheels: distance = turns × 2πr.
- % change in area = successive % change of the two lengths.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/mensuration