Mensuration: Quantitative Ability Notes for ATMA

📚 Mensuration – Quick & Complete Study Notes

(Quantitative Ability – ATMA Program)

1. Simple Explanation

Mensuration is the branch of mathematics that deals with finding the lengths, areas, volumes, and surface areas of geometric figures.

Length – 1‑dimensional measure (e.g., a line segment).
Area – 2‑dimensional measure (e.g., a square or circle).
Volume – 3‑dimensional measure (e.g., a cube or sphere).

The key to success is: know the right formula, plug in the numbers, and simplify.

2. Key Points & Formulas

Dimension	Figure	Key Formula
Length	–	`L = distance`
Area	Square	`A = a²`
	Rectangle	`A = l × w`
	Triangle	`A = ½ × base × height`
	Parallelogram	`A = base × height`
	Trapezium (US: trapezoid)	`A = ½ × (a + b) × h`
	Circle	`A = πr²`
	Ellipse	`A = πab`
Perimeter / Circumference	Square	`P = 4a`
	Rectangle	`P = 2(l + w)`
	Triangle (given sides)	`P = a + b + c`
	Circle	`C = 2πr = πd`
	Ellipse (approx.)	`C ≈ π [ 3(a + b) – √((3a + b)(a + 3b)) ]`
Volume	Cuboid	`V = l × w × h`
	Cube	`V = a³`
	Cylinder	`V = πr²h`
	Sphere	`V = 4/3 πr³`
	Cone	`V = 1/3 πr²h`
	Pyramid (regular)	`V = 1/3 × Base Area × Height`
Surface Area	Cube	`SA = 6a²`
	Cylinder	`SA = 2πr(r + h)`
	Sphere	`SA = 4πr²`
	Cone	`SA = πr(r + l)` (l = slant height)
	Pyramid	`SA = Base Area + ½ × Perimeter × Apothem`

Tip 1: For regular polygons inside a circle, use Area = ½ × Perimeter × Apothem.
Tip 2: In a circle, always remember π ≈ 22/7 or 3.1416 depending on the context.

3. Worked Example

Problem

*A right circular cone has a radius of 6 cm and a height of 8 cm.

Find the volume of the cone.
Find the slant height and the total surface area.*

Solution

1. Volume

Use ( V = \frac{1}{3}\pi r^2 h ).

[ \begin{aligned} r &= 6 \text{ cm} \ h &= 8 \text{ cm} \ V &= \frac{1}{3}\pi(6^2)(8) \ &= \frac{1}{3}\pi(36)(8) \ &= \frac{1}{3}\pi(288) \ &= 96\pi \text{ cm}^3 \ &\approx 96 \times 3.1416 \ &\approx 301.59 \text{ cm}^3 \end{aligned} ]

2. Slant Height (l)

Use the Pythagorean theorem in the right triangle formed by radius, height, and slant height:

[ l = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} ]

3. Total Surface Area

Formula: ( SA = \pi r (r + l) ).

[ \begin{aligned} SA &= \pi \times 6 \times (6 + 10) \ &= \pi \times 6 \times 16 \ &= 96\pi \text{ cm}^2 \ &\approx 96 \times 3.1416 \ &\approx 301.59 \text{ cm}^2 \end{aligned} ]

Answer:

Volume ≈ 301.59 cm³
Slant height = 10 cm
Surface area ≈ 301.59 cm²

4. Quick Revision Summary

Concept	What to Remember
Area of a circle	(A = πr²)
Perimeter of a circle	(C = 2πr)
Area of a rectangle	(A = l × w)
Area of a triangle	(A = ½ × base × height)
Area of a trapezium	(A = ½ × (a + b) × h)
Volume of a cylinder	(V = πr²h)
Volume of a cone	(V = 1/3 πr²h)
Volume of a sphere	(V = 4/3 πr³)
Surface area of a cone	(SA = πr(r + l))
Surface area of a sphere	(SA = 4πr²)
Surface area of a cube	(SA = 6