# Math Help - Area

1. ## Area

What might be the dimensions 0f a pentagonal prism if its surface area is 180cm

2. The surface area of a regular pentagonal prism is given by

$TSA = 2A_{\textrm{pentagon}} + 5lh$, where $l$ is one of the side lengths and $h$ is the height.

The area of a regular polygon is given by

$A = \frac{l^2n}{4\tan{\frac{\pi}{n}}}$, where $n$ is the number of sides, so the pentagon's area is

$\frac{5l^2}{4\tan{\frac{\pi}{5}}}$

$= \frac{5l^2}{4\sqrt{5-2\sqrt{5}}}$.

Therefore

$TSA = 2\left(\frac{5l^2}{4\sqrt{5-2\sqrt{5}}}\right) + 5lh$

$= \frac{5l^2}{2\sqrt{5 - 2\sqrt{5}}} + 5lh$.

Choose some values of $l$ and $h$ which will make this value $180\,\textrm{cm}^2$.