Find the area of the shaded region below. All shapes in the figures are regular polygons.

PLSS HELP

Step-by-step explanation:

We have 2 hexagons there, so we need to use the area formula for the large hexagon - the area of the small hexagon. As long as both of these are regular (and they are), we will use the formula

where a is the apothem and p is the perimeter of the hexagon.  We'll start with the large one first.

Note that a hexagon has 6 central angles that are all equivalent in degree.  360 / 6 = 60 degrees. Each of those 6 triangles has a vertex angle of 60.  If we extract one of those triangles and then cut it in half by dropping an altitude from the vertex angle to the base of the triangle, we get a right triangle.  The vertex angle is 30, one base angle is 90 and the other is 60. The height of this right triangle is the apothem, for which we want to solve. Using right triangle trig,

The 7 comes from the fact that when we split the triangle in half by dropping an altitude, we not only split the vertex angle in half, we also split the base length in half...from 14 to 7. Continuing,

7 tan(6) = a and

a = 12.124

Filling in the area now for the large hexagon:

{the perimeter is 6 sides times a length of 4 per side} and

A = 509.208

Now for the small hexagon. We'll do the same thing...extract a triangle and cut it in half by dropping an altitude and solving for a which is the height of the triangle. This half triangle has a base of 3 (half of 6).

3 tan(60) = a and

a = 5.196

Filling in the area now for the small hexagon:

and

A = 93.528

Now subtract the smaller area from the larger:

509.208 - 93.528 = 415.68