A geometry problem by Razing Thunder

I think you mean 36 degrees in the first line.

Since the sum of the exterior angle is 360 degrees, this is a regular 10-gon.

Let $AB$ be a side of the 10-gon, $AC$ be one of the longest diagonal of the 10-gon.

$\angle ACB=\frac12 \widehat{AB}=18^{\circ}$ , so $AB=\sin18^{\circ}AC=10\sin18^{\circ}$ ( $\angle ABC$ is a right angle.)

Hence the perimeter is $100\sin18^{\circ}$