Constructible regular polygons with ruler and compass (without computer)

Gauss's Theorem .-

A regular polygon is constructible with rule and compass if and only if $\phi(n)$ is a power of $2$ . ( $\phi$ denotes Euler's totient function )

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$\phi (3) = 2\Rightarrow$ equilateral triangle is constructible with rule and compass.

$\phi(4) = 2\Rightarrow$ square is constructible with rule and compass.

$\phi(5) = 4 = 2^2 \Rightarrow$ regular pentagon is constructible with rule and compass.

$\phi (6) = 2 \Rightarrow$ regular hexagon is constructible with rule and compass.

$\phi(7) = 6 \Rightarrow$ regular heptagon is not constructible with rule and compass. This is the smallest regular polygon not contructible with rule and compass.

$\phi(8) = 4 = 2^2 \Rightarrow$ regular octagon is constructible with rule and compass.

$\phi(9) = 6 \Rightarrow$ regular eneagon is not constructible with rule and compass. This is the second smallest regular polygon not constructible with rule and compass.

$\phi(10) = 4 = 2^2 \Rightarrow$ regular decagon is constructible with rule and compass.

$\phi(11) = 10 \Rightarrow$ regular endecagon is the third smallest regular polygon not constructible with rule and compass.