Find the area of a segment formed by a chord 8" long in a circle with radius of 8".

Answer 1

as we all know, the side of an inscribed hexagon is equal to the radius.

So, the angle subtending the side is π/3

a = 1/2 r^2 (θ - sinθ)
= 1/2 * 64 (π/3 - √3/2)
= 32/6 (2π - 3√3)

More generally, the angle θ which subtends a chord of length s in a circle of radius r is

sin θ/2 = (s/2)/r = s/2r

In this problem, s=r=8, so

sin θ/2 = 1/2
θ = π/3

and away we go.