# Regular Heptagon is Smallest with no Compass and Straightedge Construction/Proof 2

## Theorem

The regular heptagon is the smallest regular polygon (smallest in the sense of having fewest sides) that cannot be constructed using a compass and straightedge construction.

## Proof

This article needs to be tidied.Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Tidy}}` from the code. |

A theorem states that "The n-gon is constructible by compass and straightedge construction if and only if $n = 2^kn_0$, with $k \in \mathbb{Z}_{\geq0}$ and $n_0$ the product of any number of distinct Fermat primes. Note that 3 and 5 are both Fermat primes so their respective n-gon are both constructible; and $4$ is just a power of $2$ so the 4-gon also constructible. Lastly, since 6 is the product of 2 and 3, the 6-gon is constructible as well. Hence, we remain with 7, which does not satisfy the theorem's criteria; and thus the 7-gon is the smallest regular polygon that cannot be constructed using a compass and straightedge construction.

$\blacksquare$