🔢 Quant

Geometry & Mensuration

Triangles, circles, quadrilaterals, coordinate geometry, 3D solids. Typically 3–5 questions.

15%

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~15%

of section questions

Est. hours⏱️

~20h

to master

~4/22

2021

~4/22

2022

3/22

2023

~4/22

2024

3/22

2025

What CAT geometry rewards

Geometry runs to roughly 3 of the 22 Quant questions per slot. Pure plane geometry has shrunk a little, but triangles and circles remain reliable, and the questions reward recognising a configuration over heavy computation. Memorise a compact set of results and most CAT geometry becomes pattern-matching. Dedicated pages cover Triangles, Circles, Mensuration, and Coordinate Geometry in depth.

What CAT 2021–2025 actually asked

Sub-skill	2021	2022	2023	2024	2025	Avg/slot
Triangles	0.7	1.3	0.7	1.0	1.0	0.9
Circles	0.3	0.7	1.3	1.0	0.7	0.8
Mensuration (solids)	0.7	0.7	0.3	0.7	0.7	0.6
Coordinate geometry	–	0.3	0.3	0.7	0.3	0.3
Quadrilaterals & polygons	0.7	0.3	0.3	–	0.3	0.3

🎯PYQ Evidence

Triangles, circles and mensuration have appeared every single year — that trio is ~73% of CAT geometry, and it's where your prep hours belong. Coordinate geometry is newer but persistent: absent in 2021, it has appeared every year since 2022, typically as an area of a region question (2024 ran a "region defined by inequalities in the XY-plane" question in two different slots). Know the four coordinate formulas below and the region-sketching habit.

Triangles — the workhorse

Area: $\dfrac12\,b\,h$ , or Heron's $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\dfrac{a+b+c}{2}$ .
Pythagoras: $a^2+b^2=c^2$ . Know the triples on sight: 3-4-5, 5-12-13, 8-15-17 (and their multiples).
Special angles: a 45-45-90 has sides $1:1:\sqrt2$ ; a 30-60-90 has $1:\sqrt3:2$ .
Similarity: equal angles ⇒ sides in proportion, and areas scale as the square of the ratio: $\dfrac{A_1}{A_2}=\left(\dfrac{k_1}{k_2}\right)^2$ .
Inradius of a right triangle: $r=\dfrac{a+b-c}{2}$ ; in general $r=\dfrac{\text{Area}}{s}$ .

Circles — the theorems that recur

Area $=\pi r^2$ , circumference $=2\pi r$ .
The angle in a semicircle is $90°$ ; the angle at the centre is twice the angle at the circumference on the same arc.
A tangent is perpendicular to the radius at the point of contact; the two tangents from an external point are equal.

A worked example

✏️Worked Example

A right triangle has legs 6 and 8. Find its hypotenuse, area, and the radius of its inscribed circle.

Hypotenuse: $\sqrt{6^2+8^2}=\sqrt{100}=10$ (it's a $3\text{-}4\text{-}5$ scaled by 2).
Area: $\dfrac12\cdot6\cdot8=24$ .
Inradius: $r=\dfrac{a+b-c}{2}=\dfrac{6+8-10}{2}=\mathbf{2}$ . Cross-check: $r=\dfrac{\text{Area}}{s}=\dfrac{24}{12}=2.$ ✓

Coordinate geometry — the four you must know

Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Midpoint / section: the point dividing in ratio $m:n$ is $\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$
Line: $y=mx+c$ ; parallel ⇒ equal slopes, perpendicular ⇒ slopes multiply to $-1$ .
Area from vertices: $\dfrac12\,|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$ .

⚠️CAT Trap

Lengths and areas scale differently. When two similar figures have sides in ratio $k$ , their areas are in ratio $k^2$ — applying $k$ to an area (or $k^2$ to a length) is the most reliably-punished slip in CAT geometry. The same discipline applies in mensuration: doubling a cylinder's radius quadruples its volume's $r^2$ factor, while doubling its height only doubles it.

Watch this

2IIM's geometry blitzkrieg — every CAT geometry question across several years, solved:

🎯PYQ Evidence

Geometry is mostly recalling the one right relation for the figure. : the inradius formula r = (leg + leg − hypotenuse)/2 with r = 4 and one leg 10 gives the other leg, hence area 120 in one stroke. : the identity (l+b+h)² = (l²+b²+h²) + 2(lb+bh+lh) turns the edge-sum (36) and face-sum (423) into the space diagonal, which is the sphere's diameter. : angles on the same arc make the diagonal triangles similar, so AE:CE comes straight from the given side ratios AB:CD and BC:AD. Identify the governing property — inradius, diagonal-as-diameter, or equal arcs — and the computation is short.

Checklist

Spot Pythagorean triples and 30-60-90 / 45-45-90 instantly
Use area ratio = (side ratio)² for similar figures
Recall the semicircle = 90° and tangent ⊥ radius theorems
Keep the inradius formulas handy
For coordinate questions, sketch the region before computing
Draw the figure to scale — a good diagram often is the solution

Sample Questions

60 practice questions

Hard

The measures of the interior angles in a polygon are consecutive integers. The smallest angle measures 136 degrees. How many sides does this polygon have?

Easy

If x represents the sum of the interior angles of a regular hexagon and y represents the sum of the interior angles of a regular pentagon, then the difference between x and y is equal to the sum of the interior angles of what geometric shape?

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 1

Hard

In a circle with center C and radius 6 $\sqrt{2}$ cm, PQ and SR are two parallel chords separated by one of the diameters. If ∠PQC = 45°, and the ratio of the perpendicular distance of PQ and SR from C is 3:2, then the area, in sq. cm, of the quadrilateral PQRS is

CAT 2024 · Slot 1

Hard

In the XY-plane, the area, in sq. units, of the region defined by the inequalities y ≥ x + 4 and –4 ≤ $x^{2}$ + $y^{2}$ + 4(x – y) ≤ 0 is

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