How to Construct a Pentagon

If one wants to make a pentagonal frame it may help to construct a pentagon on paper using a compass and straightedge. For a pentagon the vertices are points on a unit circle separated by θ = 360°/5 = 72° but this angle is not easily found without a protractor. One can use complex numbers to find the projection of this angle, cos θ, onto the horizontal axis. Squaring the complex number, which is equivalent to doubling the angle, produces a length which is easier to construct.

One can find one half the square root of 5 since it is the hypotenuse of a right triangle with sides 1 and 1/2. This length is added to one half the radius of the circle to find cos 2θ. Drawing a perpendicular at half this distance from the center gives two of the points of the pentagon at angles 2θ and 3θ. Bisecting this the angles between these points and the horizontal axis gives the points at angles θ and 4θ. The final point is at angle θ = 0°.

The image above shows the actual construction with additional information to make it easier to follow the construction.