Deriving the area formula for a regular 5-pointed star

A regular \(5\)-pointed star can be decomposed into 5 congruent kites. Our goal is to find the area of each kite, which will then tell us the area of the star itself. Start by drawing the star's circumcircle. Draw a radius vertically from the center of the star to its uppermost vertex, and label it \(r.\) This radius seems useful, because it's both the radius of the circumcircle, which describes the size of the star, and it's one of the diagonals of the uppermost kite. Now our goal is to determine the area of the uppermost kite, in terms of \(r.\) Outline the remainder of this uppermost kite, to give it focus.

We know that the area of a kite can be determined from its diagonals, and we already have one of the diagonals, so let's draw in the other diagonal. Notice that we now have four right triangles, and all the angles within each can be determined. Label the longest leg of the large right triangle \(a,\) and the shorter leg \(b.\) Notice the shorter leg of the largest right triangle is also the shorter leg of the smallest right triangle. Label the longest leg of the smallest right triangle as \(c.\) We know that \(r = a + c,\) so the area of one kite will be \(rb,\) and thus the area of the star will be \(5rb.\) But recall that we must write everything in terms of \(r,\) and \(b\) has yet to be written in terms of \(r.\) Here's how that's done, and the final solution: