# Regular polygon A rather interesting straightedge-and-compass construction of a regular pentagon (shown in yellow)

A regular polygon is a polygon where all the sides and included angles are equal. Examples include the equilateral triangle and the square.

### Construction of Regular Polygons

The problem of which regular polygons can be constructed by ruler and compass alone goes back to the ancient Greeks. Some regular polygons (e.g. a regular pentagon) can be constructed with a straightedge (without any markings) and compass, others cannot. Carl Friedrich Gauss made the first new progress on the problem when he constructed the regular 17-gon in 1796. He later showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes .

In order to explain Gauss's Theorem, we need to understand about Fermat numbers, which are defined as numbers of the form 22n+1,

where n is natural number or zero. The first 5 Fermat numbers are: 3, 5, 17, 257, 65537 .

Gauss' Theorem on the constructibility of regular polygons says that a regular n-gon is constructible by straightedge and compass alone if and only if n = 2kp1...pt , where k and t are nonnegative integers and pi are distinct prime Fermat numbers.

By this theorem, we can construct all of the first 8 regular polygons from an equilateral triangle up to a decagon, with the exception of a heptagon. As 7 is not a Fermat prime nor has it any factors which are Fermat primes, a regular heptagon therefore cannot be constructed by straightedge and compass alone.

The construction of the regular 17-sided polygon is an inscription on Gauss' tomb.