Question

Find the area of a regular hexagon inscribed in a circle if r=20 in?

Answer 1

The regular hexagon is six equilateral triangles of side `r`, so area

`6( \sqrt{3}/{4} r^2) = {3 sqrt{3}}/2 r^2 = {3\sqrt{3}}/2 (20^2)=600 sqrt{3}`

Explanation:

The regular hexagon is six equilateral triangles, the sides of which in this problem are `r`. The area of one equilateral triangle is `\sqrt{3}/{4} r^2` so six of them have area `{3sqrt{3}}/{2} r^2`.

Let's prove the area of an equilateral triangle a couple of different ways. An altitude of height `h` bisects one side `r`, giving a right triangle

`(r/2)^2 + h^2 = r^2`

`r^2/4 + h^2 = r^2`

`h^2 = 3/4 r^2`

`h = sqrt{3}/2 r`

`A = 1/2 r h = \sqrt{3}/4 r^2 quad sqrt`

That required some thinking. Another way it to remember the modern form of Heron's Formula, called Archimedes' Theorem. A triangle with sides `a,b,c` and area `A` satifies

`16A^2 = 4a^2b^2-(c^2-a^2-b^2)`

Let's set `a=b=c=r:`

`16 A^2 = 4r^4 - r^4 = 3r^4`

`A^2 = 3/16 r^4`

`A = \sqrt{3}/4 r^2`

Much easier.