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.
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.