Question

What is the area of a regular hexagon with apothem 7.5 inches? What is its perimeter?

Answer 1

A hexagon can be split up into 6 equilateral triangles.
If one of these triangles has a height of 7.5 in, then (using the properties of 30-60-90 triangles, one side of the triangle is `(2*7.5) / sqrt3= 15/sqrt3 = (15sqrt3)/3`.
Since the area of a triangle is `(1/2)*b*h`, then the triangle's area is `(1/2)(15sqrt3/3)*(7.5)`, or `(112.5sqrt3)/6`. There are 6 of these triangles that make up the hexagon, so the area of the hexagon is`112.5*sqrt3`.

For the perimeter, again, you found one side of the triangle to be `(15sqrt3)/3`.
This is also the side of the hexagon, so multiply this number by 6.