Find the perimeter

Question

A regular octagon is inscribed in a circle with a radius of 7 inches. Find the perimeter of the octagon. Round to the nearest tenth.

Answer 1

The perimeter of an inscribed n-sided polygon of radius r is 2nr sin π/n. Note how, as n goes to infinity, this goes to 2nr π/n = 2rπ, the perimeter of a circle of radius r.

For n = 8, sin π /8 = 1/2√(2 - √2) (You can derive this using the half-angle formula sinπ /8 = √((1-cosπ /4)/2)

2nr sin π/n = 2*8*7*sin π/8 = 56√(2 - √2) = 42.9