Question

Using the formula to find the area of a regular polygon, what is the area of a regular 15 sided polygon with side lengths of 9 millimeters and an apothem that measures 12 millimeters.

Answer 1

To find the area of a regular polygon, we can use the formula:

\[
\text{Area} = \frac{1}{2} \times \text{perimeter} \times \text{apothem}
\]

First, we need to determine the perimeter of the regular 15-sided polygon. The perimeter \( P \) can be calculated by multiplying the number of sides \( n \) by the length of each side \( s \):

\[
P = n \times s
\]

For our regular polygon, \( n = 15 \) and \( s = 9 \) mm:

\[
P = 15 \times 9 = 135 \text{ mm}
\]

Now that we have the perimeter, we can substitute \( P \) and the apothem \( a = 12 \) mm into the area formula:

\[
\text{Area} = \frac{1}{2} \times 135 \times 12
\]

Calculating that:

\[
\text{Area} = \frac{1}{2} \times 135 \times 12 = \frac{1}{2} \times 1620 = 810
\]

Thus, the area of the regular 15-sided polygon is:

\[
\boxed{810} \text{ square millimeters}
\]