Side of a regular polygon inscribed in a circle

How to use it

1. 1For the square, use that the diagonal is a diameter: $a\sqrt{2}=2R$.
2. 2For the equilateral triangle, the centre lies at distance $\frac{R}{2}$ from each side, so half the side and $\frac{R}{2}$ are the legs of a right triangle with hypotenuse $R$.
3. 3In a ratio of two inscribed figures the radius cancels, so the answer is a pure number independent of the circle.

Step-by-step

1. 1
   $a_6=2R\sin 30^{\circ }=R=10$ cm

   Why: Each side of an inscribed regular hexagon equals the radius.

2. 2
   $6\times 10=60$ cm