Area of a triangle and connected circles

Maths · Application of trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The area of a triangle is written Δ. Which formula you use depends on what is given. With two sides and the included angle, use half the product of the sides times the sine of that angle. With three sides, use Hero's formula, which needs the semi-perimeter s.

Three kinds of circle are connected with every triangle. The circum-circle passes through the three vertices; its radius is R. The in-circle lies inside and touches all three sides; its radius is r. Each escribed circle touches one side from outside and the other two sides produced; the radii are r₁, r₂, r₃, opposite A, B and C. All these radii come from Δ and s.

Rules to remember

Δ = ½ab sin γ = ½bc sin α = ½ca sin β
Semi-perimeter: s = (a + b + c)/2
Hero's formula: Δ = √(s(s − a)(s − b)(s − c))
Circum-radius: R = a/(2 sin α) = b/(2 sin β) = c/(2 sin γ) = abc/(4Δ)
In-radius: r = Δ/s
Escribed radii: r₁ = Δ/(s − a), r₂ = Δ/(s − b), r₃ = Δ/(s − c)
In a right-angled triangle the circum-radius is half the hypotenuse

Common mistake

Using the full perimeter in place of s. Always halve the perimeter first, and remember r = Δ/s but R = abc/(4Δ).

Worked example

A triangle has sides a = 9, b = 10 and c = 17. Find its area, its in-radius and its circum-radius.

s = (9 + 10 + 17)/2 = 18, so s − a = 9, s − b = 8, s − c = 1.
Δ = √(18 × 9 × 8 × 1) = √1296 = 36.
r = Δ/s = 36/18 = 2.
R = abc/(4Δ) = (9 × 10 × 17)/(4 × 36) = 1530/144 = 10.625.

Answer: Δ = 36 square units, r = 2 and R = 85/8 = 10.625.

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