mathschallenge.net

Problem

A triangle is formed by connecting the two ends of the diameter of a semi-circle, length c, to a point on the circumference. Circles are constructed on the two shorter sides with diameters a and b respectively, so as to form two lunes (the shaded part).

Find the total area of the two lunes in terms of a, b, and c.

Solution

Consider the diagram, with regions identified by the numbers 1 to 6.

In a semi-circle the triangle will be right-angled, so using the Pythagorean Theorem, a² + b² = c².

Multiplying through by π/8 we get, ½π(a/2)² + ½π(b/2)² = ½π(c/2)². In other words, if semi-circles are drawn on the sides of a right-angle triangle, the area of the semi-circles on the shorter sides will be equal to area of the semi-circle on the hypotenuse.

That is, (A₁ + A₂) + (A₃ + A₄) = A₆; in addition, A₆ = A₂ + A₄ + A₅.

A₁ + A₂ + A₃ + A₄ = A₂ + A₄ + A₅ A₁ + A₂ = A₅

Hence the area of the two lunes are equal to area of triangle, ½ab.