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 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**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*.*D'EA*
is an isosceles triangle, *m* + *n* = *s*.

There is only one solution to the quadratic equation because, in this
case, we know that *φ* must be positive.
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*.*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 *φ*^{2} = *φ* + 1.

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

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

Last update: November 2, 2011 ... Paul Kunkel

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