Side of an inscribed regular polygon

Problem

A regular polygon has $n$ sides inscribed in a circle of radius $R$. Determine the length of the side as a function of $n$ and $R$.

Solution

Every side of the polygon is a chord of the circle. Joining two consecutive vertices to the centre $O$ forms an isosceles triangle with the two equal sides equal to $R$ and central angle $\dfrac{2\pi}{n}$ (the full angle divided by the number of sides).

Applying the chord theorem, the inscribed angle subtending the side is half the central angle, that is $\dfrac{\pi}{n}$, so the side $\ell$ is: $\ell = 2R\sin\frac{\pi}{n}.$ (Equivalently, in the isosceles triangle drop the bisector from the central angle, obtaining two right-angled triangles with opposite side $\ell/2$ and hypotenuse $R$: $\ell/2 = R\sin\tfrac{\pi}{n}$.) $\boxed{\ell = 2R\sin\frac{\pi}{n}}$

Links

Topics: Triangle trigonometry
Concepts: Regular polygon · Chord theorem
Functions: Sine
Skills: Synthetic geometry · Modelling · Using formulae
Exercise type: Geometric problem