Properties of Triangles: JEE Main Complete Guide

Properties of triangles is a dedicated JEE Main topic that consistently yields one to two questions per session. It requires knowledge of the sine rule, cosine rule, area formulas, circumradius, inradius, and the half-angle formulas. The chapter is formula-dense but the formulas are interconnected — once you know the sine rule and the area formula, most other results follow. This guide covers every formula and question type JEE Main uses.

Sine Rule and Cosine Rule

Sine rule: a/sinA = b/sinB = c/sinC = 2R, where a, b, c are the side lengths opposite to angles A, B, C respectively, and R is the circumradius. This means: R = a/(2sinA) = b/(2sinB) = c/(2sinC). JEE Main uses the sine rule in two ways: (1) finding an unknown side or angle given others; (2) proving identities involving a/sinA combinations.

Cosine rule: a² = b² + c² − 2bc·cosA. Rearranged: cosA = (b² + c² − a²)/(2bc). The cosine rule is the general form of the Pythagorean theorem (reduces to it when A = 90°). JEE Main uses cosine rule for: (1) finding the third side given two sides and the included angle; (2) finding an angle given three sides. For trigonometry fundamentals, see our Trigonometry Guide.

Area Formulas

Area Δ = ½bc·sinA = ½ac·sinB = ½ab·sinC (half product of two sides and sine of included angle). Heron's formula: Δ = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. Relation to inradius r: Δ = rs (inradius × semi-perimeter). Relation to circumradius R: Δ = abc/(4R). JEE Main tests these four area expressions and asks you to relate them — for example, prove that r = Δ/s, or find R given a, b, c. Take a free mock test on trigonometry to practise sine rule, cosine rule, and area problems.

Circumradius and Inradius

Circumradius R = a/(2sinA) = abc/(4Δ). Inradius r = Δ/s = (s−a)tanA/2 = 4R·sinA/2·sinB/2·sinC/2. The inradius formula r = (s−a)tan(A/2) is frequently tested — it directly connects r to the half-angle and semi-perimeter. For an equilateral triangle of side a: R = a/√3, r = a/(2√3), and r/R = 1/2. JEE Main uses the r/R ratio for equilateral triangles as a standard result.

Half-Angle Formulas for Triangles

sin(A/2) = √[(s−b)(s−c)/(bc)]. cos(A/2) = √[s(s−a)/(bc)]. tan(A/2) = √[(s−b)(s−c)/(s(s−a))] = r/(s−a). These half-angle formulas in triangles are standard JEE Main results. They appear in proofs and in computation problems where you are asked to find sin(A/2) given the side lengths. The tan(A/2) = r/(s−a) formula directly connects half-angles to the inradius.

Excircles

An excircle opposite vertex A has radius r_A = Δ/(s−a) = s·tan(A/2). Similarly, r_B = Δ/(s−b) and r_C = Δ/(s−c). Important relation: 1/r = 1/r_A + 1/r_B + 1/r_C (the sum of reciprocals of excircle radii equals the reciprocal of inradius). JEE Main occasionally asks for excircle radius given the triangle dimensions — use r_A = Δ/(s−a) directly.

Important Identities

a = b·cosC + c·cosB (projection formula). a·sinB = b·sinA (from sine rule). (b+c)/(a) = cos((B−C)/2)/sin(A/2) (Napier's analogy). A+B+C = π → tanA + tanB + tanC = tanA·tanB·tanC (if A+B+C = π). JEE Main tests Napier's analogy and the product formula for tan in proofs and computation. For coordinate geometry applications of triangles, see our Coordinate Geometry Guide.

Exam Strategy

Properties of triangles questions usually require selecting the right formula from a list of closely related ones. The key is knowing what each formula computes: sine rule for ratios of sides to sines; cosine rule for side-angle calculation; area formulas for Δ; r = Δ/s for inradius; R = abc/(4Δ) for circumradius. Practise substituting into these formulas with specific numbers to build familiarity with their outputs. Upgrade for ₹149/month for 100+ triangle properties problems with complete step-by-step solutions.