The Regular Pentagon

The regular pentagon is a deep file. Within it can be found self-similarities and repeated occurances of the golden ratio.

Begin with regular pentagon ABCDE, with side length s. By drawing the five diagonals of that figure, we can inscribe a certain star commonly called a pentagram. Let A', B', C', D', and E' be the five points of intersection of the diagonals.

From the symmetry of the figure, it is clear that C'AD' is an isosceles triangle. If we circumscribe a circle about the pentagon, we see that ∠AEB = ∠DAC because they are circumferal angles intercepting equal chords. It follows that triangles D'EA and C'AD' are similar. Isolate those two triangles, and other interesting relations appear.

From the symmetry of the pentagram, EC' = AD'. Let that length equal m, and let C'D' = n. Since D'EA is an isosceles triangle, m + n = s. Corresponding sides of similar figures have equal ratios. Therefore:

There is only one solution to the quadratic equation because φ is the ratio of two line segment lengths, and therefore cannot be negative. This ratio makes many appearances in the pentagon. Notice above that triangle DAC is similar to triangle C'AD'. Therefore, AC/CD = AD'/D'C' = φ. Line segment AC is a diagonal of the pentagon. What was shown here is that for a regular pentagon, φ is the ratio of a diagonal to a side.

Now that we know the ratio of a diagonal to a side, that ratio can be used to evaluate several trigonometric functions of angles found in the pentagon. These values will be needed in later calculations. Consider a pentagon with unit side length. The measure of its diagonal must then be φ. Cut the pentagon at a diagonal to form triangle STU, as shown below. All of the sides and angles are known.

In calculations involving φ, it is often helpful to make use of the identity φ² = φ + 1. This comes directly from the quadratic equation above. Watch for it in the following calculations.

For some of the dihedral angle calculations, it will be necessary to know cos(3π/5) and sin(3π/5). Use the law of cosines for this.

For the pentagon apothem, tan(π/5) will be useful. This calculation might be simplified by constructing the midpoint, M, of side SU. This results in right triangle SMT.