In an isosceles triangle, if the measure of the leg angle is 72°, then the base and a leg are incommensurable. (Two lengths

and

are called incommensurable if their ratio is irrational.)

Let the vertices of the triangle be

, where

is the base. The angles are 72°, 72°, and 36°.

Assume that the ratio of

to

is rational, say

for positive integers

and

. For some appropriate unit of measurement, we can take the lengths of

and

to be

and

, where

.

Bisect the angle at

to meet the opposite side at

. The triangle

is an isosceles triangle because the angles are 36°, 36°, and 108°. The triangle

is an isosceles triangle because its angles are 72°, 72°, and 36° (

is similar to the original triangle

). Therefore,

and

.

Iterating this construction gives the points

,

, … and

,

, … with

,

, …, where each of the integers is positive and forms a strictly decreasing sequence, which is impossible.

Therefore, the ratio of the sides is irrational.

In fact, the ratio is

, the golden ratio. Its continued fraction is

with cumulants 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, …; you can see the numerators and denominators in the table.