Derivation of Area of Regular Pentagon Formula

So, when I take one of my children to his parkour lesson (yes, you read that right), I often sit in the car waiting for him to finish (i’m not allowed to watch!). To the amusement of many, I have an emergency maths kit for such occasions (notebook, pen, pencil, ruler, big circular coin – for … circles). All very geeky indeed. Anyway, I thought I’d see if I could derive the formula for the area of a regular pentagon, or at least, an equivalent one.

The area formulae of regular odd-gons (2n-1 gons?) are always ugly, so I figured this would  be a big mess, but after having an afternoon of being TERRIBLE at maths yesterday, I thought I needed the activity to get back on track. I think the notes are fairly straight forward to follow. I basically split the shape into two identical triangles, and a remaining isosceles, and just used the side length, rather than the apothem. I particularly enjoyed how the nasty surd mess in the middle became really nice after a hard slog attempt at simplifying it. (click images for enlargement)



My final formula is at the bottom of the second page, and it’s pretty similar to the one you’ll find elsewhere:


In my thinky state, I also noticed a couple of cool things about pentagons:



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