Question

What is the area of this hexagon?

Answer 1

Most likely, this question is about a regular hexagon with a given size `a` of a side.

The answer is:
`S = (3sqrt(3))/2a^2`

Explanation:

If we connect a center of a regular hexagon with all `6` vertices, we will get `6` equilateral triangles with each side equal to `a`.

The proof is trivial. Since this hexagon is regular, the distances from its center to all vertices are equal to each other. Hence, all these triangles are isosceles with a top being in the center of a hexagon.
In addition, an angle at the top of each triangle is `1/6` of `360^o`, that is `60^o`.
Two other angles of each triangle, therefore, are also equal to `60^o`.
That makes each isosceles triangle an equilateral one.

An equilateral triangle with a side `a` has an altitude equal to
`sqrt(a^2-(a/2)^2) = a sqrt(3)/2`

Therefore, its area is
`S_(Delta) = 1/2 a^2 sqrt(3)/2 = sqrt(3)/4a^2`

The area of a hexagon in `6` times greater, that is
`S_(hexagon) = (6sqrt(3))/4a^2 = (3sqrt(3))/2a^2`