The area of a regular polygon is half the product of the apothem and the perimeter. Note that we are given the apothem but are missing the perimeter. Let's first find the perimeter and use it to find the area.

Finding the Perimeter

To find the perimeter, let's start by drawing the radii of the given polygon.

The radii divide the square into four congruent isosceles triangles. Since corresponding angles of congruent figures are congruent, the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures

36 0^{\circ},

we can divide

360

by

4

to obtain their measures.

\frac{3 6 0}{4} = 9 0^{\circ}

The vertex angles of the isosceles triangles measure

9 0^{\circ}

each.

Next, recall that the apothem bisects the vertex angle of the isosceles triangle formed by the radii. As a result, $4 5^{\circ} - 4 5^{\circ} - 9 0^{\circ}$ triangles are obtained. Let's consider one of them.

In this type of special triangle, the legs are congruent and length of the hypotenuse is

2

times the length of a leg.

Other Leg : 5 in .

Not only does the apothem bisect the vertex angle of the right triangle but it also bisects its opposite side, which is a side of the square. Therefore, the length of one side of the given regular polygon is

Exercise