Pentadecagon

Pentadecagon

[pen-tuh-dek-uh-gon]
In geometry, a pentadecagon is any 15-sided, 15-angled, polygon. A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by
begin{align} A & = frac{15}{4}a^2 cot frac{pi}{15} & = frac{15a^2}{8} left(sqrt{3}+sqrt{15}+ sqrt{2}sqrt{5+sqrt{5}}
                                    right) 
                & simeq 17.6424,a^2.
end{align}


Pentadecagon construction

A regular pentadecagon is constructible with compass and straightedge. The following animation illustrates this process in 36 steps, and is adapted from a method given in Euclid's Elements, Book IV, Proposition 27. Note that the compass radius remains unaltered during steps 14 through 21.

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