Question

What is the area of a regular hexagon circumscribed iinside a circle with a radius of 1?

Answer 1

`frac{3sqrt{3}}{2}`

Explanation:

The regular hexagon can be cut into 6 pieces of equilateral triangles with length of 1 unit each.

For each triangle, you can compute the area using either

1) Heron's formula, `"Area"=sqrt{s(s-a)(s-b)(s-c)`, where `s=3/2` is half the perimeter of the triangle, and `a`, `b`, `c` are the length of the sides of the triangles (all 1 in this case). So `"Area"=sqrt{(3/2)(1/2)(1/2)(1/2)}=sqrt{3}/4`

2) Cutting the triangle in half and applying Pythagoras Theorem to determine the height (`sqrt{3}/2`), and then use `"Area"=1/2*"Base"*"Height"`

3) `"Area"=1/2 a b sinC=1/2 (1) (1) sin(pi/3)=sqrt{3}/4`.

The area of the hexagon is 6 times the area of the triangle which is `frac{3sqrt{3}}{2}`.