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Triangles for CAT: Formulas, Weightage & Question Types

Right triangles with altitudes and incircles, centroids, area splits and hidden similarity.

Triangles formula sheet

Everything CAT needs from Triangles, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place. aa, bb, cc are the sides, s=a+b+c2s = \dfrac{a+b+c}{2}.

Area, four ways

Base and height12×base×height\dfrac{1}{2} \times \text{base} \times \text{height}
Herons(s−a)(s−b)(s−c)\sqrt{s(s-a)(s-b)(s-c)}
Two sides and the angle between12absin⁡C\dfrac{1}{2}ab\sin C
Equilateral, side aa34a2\dfrac{\sqrt{3}}{4}a^2

Right-angled triangles

Pythagorasa2+b2=c2a^2+b^2=c^2
Altitude to the hypotenuseabc\dfrac{ab}{c}
Largest that altitude can bec2\dfrac{c}{2}, when the triangle is isosceles
Inradiusr=a+b−c2r = \dfrac{a+b-c}{2}
CircumradiusR=c2R = \dfrac{c}{2} — the hypotenuse is a diameter
Triples worth knowing3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 20-21-29

Centres

Centroidwhere the medians meet, splitting each 2 : 1 from the vertex
One mediancuts the triangle into two equal areas
All three medianscut it into six equal areas
Two medians perpendicular, lengths m1,m2m_1, m_2area =23m1m2= \dfrac{2}{3}m_1m_2
Inradiusr=areasr = \dfrac{\text{area}}{s}
CircumradiusR=abc4×areaR = \dfrac{abc}{4 \times \text{area}}

Ratios of areas

Similar trianglesareas are in the square of the side ratio
DD on ABAB and EE on ACAC[ADE][ABC]=ADAB×AEAC\dfrac{[ADE]}{[ABC]} = \dfrac{AD}{AB} \times \dfrac{AE}{AC}
Same height, different basesareas are in the ratio of the bases
Midlinejoins two midpoints, parallel to the third side and half its length
Medial trianglearea is 14\dfrac{1}{4} of the original

Angles

Sine ruleasin⁡A=bsin⁡B=csin⁡C=2R\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R
Cosine rulec2=a2+b2−2abcos⁡Cc^2 = a^2+b^2-2ab\cos C
A side that is a diameterthe angle facing it is 90∘90^\circ
Exterior angleequals the sum of the two opposite interior angles

Two results that answer whole questions

Vivianiin an equilateral triangle, the perpendiculars from any interior point to the three sides always add to the altitude
Triangle inequalityeach side is less than the sum of the other two

The 6 Triangles question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine.

  1. 01

    A right triangle, with an altitude or an incircle

  2. 02

    Medians and the centroid

  3. 03

    A point on a side, and an area split

  4. 04

    Equilateral or isosceles, with perpendiculars

  5. 05

    Chasing angles

  6. 06

    Similar triangles, often hidden

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Triangles in CAT: quick answers

How many Triangles questions come in CAT Quant?
About 1 question in a 22-question Quant section, worth roughly 3 marks. Triangles appeared in 17 of the 24 CAT papers from 2017 to 2025, 22 questions in all, which ranks it 10 of 25 Quant chapters.
Are Triangles questions in CAT MCQ or TITA?
46% of the Triangles questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Triangles questions does CAT ask?
CAT keeps returning to 6 question shapes in Triangles: A right triangle, with an altitude or an incircle; Medians and the centroid; A point on a side, and an area split; Equilateral or isosceles, with perpendiculars; Chasing angles; Similar triangles, often hidden.

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