Find the area of a regular octagon inscribed in a unit circle?

Think of the octagon as eight triangles put together. The corners of the octagon are touching the circumference of the circle, so the distance between the center of the octagon and the corners is one. Now you have eight isosceles triangles. The top angle of the triangles is 360 divided by 8 which is 45, and the sides around it are 1 and 1. With this information, the bases of the triangles can be calculated to a value of 2/3, or .6667. The height of the triangles is .9428. The area of each triangle is (1/2)bh, or (.5)(.6667)(.9428) which is equal to .3143. Now multiply it by eight, and you have the total area of the octagon (2.5142).

The area of the octagon is approximately 2.828, because the area of a polygon greater than a square is 1/2 the apothum times the perimeter. The apothum is the line segment from the center point of the polygon perpendicular to a side. So, we know the distance from the center to a vertex, which is one, because it is a unti circle. So we draw a line from a vertex to the center and an apothum. The angle formed between the two is one half the measure of the interior angle of the octogon which is 360 divided by 8. You use trig to find the measure of one half of the side you're using, double it, and multiply by eight to find the perimeter. Now you have all the elements to plug it into the equation, 1/2 apothum times the perimeter,