Mensuration: Concepts, Formulas, and Clear Examples

Mensuration is 2D/3D measurement: perimeter, area, surface area, volume. Aptitude favors cylinder, cone, sphere, cuboid and composite figures split into standard shapes.

Keep units consistent (cm vs m); converting dimensions scales area by $k^2$ and volume by $k^3$.

For frustums and hemispheres, write formula first, then substitute—avoid mixing diameter and radius.

2D basics

Rectangle

$$A = l\times b,\ P = 2(l+b)$$

Circle

$$A=\pi r^2,\ C=2\pi r$$

Triangle

$$A=\frac{1}{2}bh$$

Cuboid and cube

Cuboid volume

$$V = l\times b\times h$$

Cuboid TSA

$$2(lb+bh+hl)$$

Cube

$$V=a^3,\ \text{TSA}=6a^2$$

Cylinder

Volume

$$V = \pi r^2 h$$

Curved SA

$$2\pi r h$$

Total SA

$$2\pi r(h+r)$$

Example — Cylinder volume

$r=7\text{ cm}$, $h=10\text{ cm}$. $V=\pi\cdot49\cdot10=490\pi\text{ cm}^3$.

Cone and sphere

Cone volume

$$V=\frac{1}{3}\pi r^2 h$$

Sphere volume

$$V=\frac{4}{3}\pi r^3$$

Sphere SA

$$4\pi r^2$$

Frustum / hemisphere

Hemisphere volume

$$\frac{2}{3}\pi r^3$$

Hemisphere CSA

$$2\pi r^2$$

Composite solids

• Add volumes of joined parts; subtract cavities.
• Painted surface: count exposed faces only.

Step-by-step method for exam questions

1. List given as radius/diameter/height clearly.
2. Choose 2D vs 3D formula set.
3. Split composite into standard pieces.
4. Substitute; keep π factor if options use it.
5. Unit-check (cm³ vs m³).
6. Select answer.

Formula cheat sheet

Frequently asked questions

Diameter trap?

Halve to radius before $\pi r^2$.

CSA vs TSA?

TSA includes bases; melting/recasting uses volume only.

Hemisphere on cylinder?

Add volumes; TSA excludes flat join if internal.

Same volume different shape?

Equate volume formulas to find unknown dimension.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.