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)

IMG_0764

IMG_0765

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

pent

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

IMG_0766

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