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regular polygons
Polygon area = 1/2 perimeter x apothem
When you add up all the areas of the triangle sectors in a polygon
you end up with something pretty neat: 1/2 perimeter × apothem.
What do you think the trend is?
Which hot tub are you leaning toward?
Joe: Yeah, cool, whatever. Except we still need to find a
bunch of apothems to use it!
Frank: Relax, I’m not sure we actually do.
Jill: How come?
Frank: Well, the whole way we found that circle area
formula was about adding more and more sides to the
polygon until it resembled a circle—yeah?
Joe: Yeah. And?
Frank: So how many butts fit in the circle tub?
Jill: Not many! It’s the least we’ve found. Triangle was
the most, square was pretty good.
Frank: Exactly! That’s not just a detail—that’s a trend!
OK, great. So that gives us a
formula that we can use for all
the hot tubs, whether they’ve got
6, 7, 8, or 9 sides. That is cool.
Frank
Jill
Joe
Often you’ll be given the apothem,
or you might have the circumradius
and the side length and then you
can use the Pythagorean Theorem
to find the apothem.