Suppose you want to find the area of a dodecagon inscribed in a circle of radius 1. One way to do this is to divide the dodecagon into 12 isosceles triangles with two sides of length 1 and an angle between them of 30°. Then we can use the formula for the area of a triangle $$A = \frac12ab \sin C$$ to give us an area of $$ \frac12 \times 1^2 \times \sin 30 = \frac14$$ Multiplying by 12, the area of our dodecagon is therefore 3 square units.
There’s another way to get this result which doesn’t rely on knowing that $\sin 30 = \frac12$. I first came across this dissection of a dodecagon when working at NRICH and looking for ideas for the Wild Maths site, which was about exploring mathematical creativity. Take a look at the GeoGebra construction below.
Move the slider, or click the play button to start the animation. There’s a little bit of thinking to be done to convince yourself that the pieces fit nicely together with no gaps, but I hope you agree it’s a very pleasing way to show that the area of the dodecagon is 3 square units!