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Area of a dodecagon without trigonometry

How do you find out the area of this dodecagon without using trig?
Rearrange the triangles to make…
Three squares! The area of the dodecagon with a radius r is equal to the area of three r-sided squares or 3r^2.
The formulas for other polygons. There seem to be no formulas for the heptagon, nonagon and hendecagon (without using trigonometry that is). Simon’s notes above also say that no polygon can possibly have an area equal to or larger than πr^2 (because that’s the area of a circle). A square is 2r^2, a dodecagon is 3r^2, and so no polygon is 4r^2.

Simon learned this from a visual mathematics video by Think Twice.

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