regular decagon is inscribed in a circle with diameter of exactly 20 cm. Find the length of one side of the decagon, to the nearest tenth of a centimeter

Answer 1

The measure of one interior angle of a regular polygon = (n-2)•180° / n.

For decagon, n=10, one interior angle has a measure of:

(10-2)•180° / 10 = 144°

You can make a line segment from one corner to another passing through the center to make a diameter. There are 5 diameters total. Each diameter will cut the interior angle into two equal parts.

Therefore we can get one internal isosceles triangle which has angle measure of 36°-72°-72°,

360°/10 =36°

144°/2 = 72°

and the sides are x- 10 cm. -10 cm where x is the length of one side of decagon and the two 10 cm.'s are also the lengths of the two radii of a circle.

to solve for x, we can use the Law of Cosine:

x² = a² + b² - 2ab • cos θ

x²= 10² + 10² - 2(10)(10) • cos 36°

x ≈ 6.18 cm.