Question

Is there an algebraic formula to find out the area of a hexagon?

Answer 1

Depending upon what properties are known:

`A = 3 a h`, or `A = (3sqrt(3))/2 \ a^2`

Explanation:

Consider (as pictured:) a regular hexagon, with side length `a`. The regular hexagon is composed of `6` equilateral triangles, if we denote the height of one such triangle by `h`, the the area of a single triangle is:

`A_T = 1/2 xx "base" xx "height`
`\ \ \ \ \ = 1/2 a h`

Thus the area of the entire hexagon, is given by:

`A = 6 A_T`
`\ \ = 6/2 a h`
`\ \ = 3 a h`

If `h` is unknown, then we use pythagoras to get:

`a^2 = h^2 + (a/2)^2`
`\ \ \ = h^2 + a^2/4`

`:. h^2 = (3a^2)/4`

`h = (sqrt(3)a)/2`

Thus we can write:

`A = (3) xx (a) xx ((sqrt(3)a)/2)`
`\ \ \ = (3sqrt(3))/2 \ a^2`