Calculating the area of a polygon with equal sides inscribed in a circle with radius r.

Question

Calculating the area of a polygon with equal sides inscribed in a circle with radius r.By dividing the polygon into congruent triangles with central angle, show that: $$A_{n} = \frac{1}{2}n r^2 \sin\left(\frac{2\pi}{n}\right),$$

How can I use Darboux sums to do so, could anyone give me a hint please?

Answer 1

Can you find the area of the triangle in this diagram? Do you see how to use it to find the area of an $n$-sided regular polygon inscribed in the circle?

You will need the identity

$$\sin2\theta=2\sin\theta\cos\theta$$