🔢 Quant

Mensuration & Polygons

Areas and volumes of 2D and 3D figures, regular polygons.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~3%

of section questions

Est. hours⏱️

~6h

to master

~1/22

2021

~1/22

2022

~1/22

2023

~1/22

2024

~1/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.6 per slot (2021: 0.7 · 2022: 0.7 · 2023: 0.3 · 2024: 0.7 · 2025: 0.7). Every year (under 1 per slot); cylinders, cones and spheres in combination.

Mensuration

Area and volume of 2-D and 3-D figures. The formulas are few; the marks come from picking the right one and watching which surface is asked.

2-D areas

Rectangle $l\,b$ ; triangle $\tfrac12 b h$ ; circle $\pi r^2$ .
Trapezium $\tfrac12(a+b)h$ ; rhombus $\tfrac12 d_1 d_2$ .

3-D solids

Solid	Volume	Curved/lateral surface
Cuboid	$l\,b\,h$	$2h(l+b)$
Cylinder	$\pi r^2 h$	$2\pi r h$
Cone	$\tfrac13\pi r^2 h$	$\pi r l$ (slant $l=\sqrt{r^2+h^2}$ )
Sphere	$\tfrac43\pi r^3$	$4\pi r^2$

A worked example

A cylinder has radius 7 cm and height 10 cm. Find its volume. (Take $\pi=\tfrac{22}{7}$ .)

$V=\pi r^2 h=\frac{22}{7}\times 7^2\times 10=\frac{22}{7}\times 49\times 10=22\times 7\times 10=\mathbf{1540\ \text{cm}^3}.$

Choosing $\pi=\tfrac{22}{7}$ with a radius that's a multiple of 7 makes the arithmetic vanish — a common CAT setup.

🎯PYQ Evidence

Mensuration rewards a few structural identities over brute formula-recall. : the surface area gives lb+bh+lh = 423 and the edge sum gives l+b+h = 36, and the identity (l+b+h)² = l²+b²+h² + 2(lb+bh+lh) yields the space diagonal — which is the sphere's diameter. : with area LB = 60000 fixed, write cost = 300L + 200B and minimise by AM-GM (a+b ≥ 2√(ab)), since the product of the two terms is fixed — bottoming out at Rs 120000. : every side is less than the sum of the other three, so 1 < x < 7 forces integer x ∈ {2,3,4,5,6}, i.e. 5 values. Look for an identity, an inequality, or a polygon side-rule before computing.

Common traps

Curved vs total surface area. Total includes the flat ends ( $+2\pi r^2$ for a cylinder, $+\pi r^2$ for a cone).
Radius vs diameter. Many questions give the diameter — halve it first.
Mixed units. Convert everything to one unit before computing volume.

Checklist

Match the solid to its volume formula
Distinguish curved / lateral from total surface area
Compute the cone's slant $l=\sqrt{r^2+h^2}$ when needed
Pick $\pi=\tfrac{22}{7}$ when the radius is a multiple of 7

Sample Questions

16 practice questions

Hard

The height of isosceles trapezoid ABDC is 12 units and the length of diagonal AD is 15 units. What is the area of trapezoid ABDC?

Medium

Two black squares have a combined area of 1000 sq units. The larger side is 8 units longer than the smaller. What is the combined area of the two white rectangles formed between them?

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 2

Hard

Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively. Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is

CAT 2024 · Slot 1

Medium

The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

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