Inradius, Semiperimeter, and Area

$$\text{Pi Coincidence?}$$

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The number $\pi$ is defined to be the ratio circumference / diameter for any circle. We also all know that the area of a circle is $\pi r^2$. Is it a sheer coincidence that they are both the same $\pi$, even though one concerns the circumference and one concerns the area?

No! Let's do it for a regular pentagon. It turns out that for the appropriate definition of the "diameter" of a regular pentagon, if we define the number $\phi$ to be the ratio of the perimeter / diameter of any regular pentagon, then its area is always $\phi r^2$, where $r$ is half of the diameter.