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
Answers
Answered by
GPT-4o mini
To find the area of a regular polygon, you can use the formula:
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
First, we need to calculate the perimeter of the regular 15-sided polygon (also known as a pentadecagon). The perimeter is calculated by multiplying the number of sides by the length of each side:
\[
\text{Perimeter} = \text{Number of sides} \times \text{Side length} = 15 \times 9 \, \text{mm} = 135 \, \text{mm}
\]
Now, with the perimeter and the apothem known, we can substitute these values into the area formula:
\[
\text{Area} = \frac{1}{2} \times 135 \, \text{mm} \times 12 \, \text{mm}
\]
Calculating the area:
\[
\text{Area} = \frac{1}{2} \times 135 \times 12 = \frac{1620}{2} = 810 \, \text{mm}^2
\]
Thus, the area of the regular 15-sided polygon is:
\[
\boxed{810 \, \text{mm}^2}
\]
\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]
First, we need to calculate the perimeter of the regular 15-sided polygon (also known as a pentadecagon). The perimeter is calculated by multiplying the number of sides by the length of each side:
\[
\text{Perimeter} = \text{Number of sides} \times \text{Side length} = 15 \times 9 \, \text{mm} = 135 \, \text{mm}
\]
Now, with the perimeter and the apothem known, we can substitute these values into the area formula:
\[
\text{Area} = \frac{1}{2} \times 135 \, \text{mm} \times 12 \, \text{mm}
\]
Calculating the area:
\[
\text{Area} = \frac{1}{2} \times 135 \times 12 = \frac{1620}{2} = 810 \, \text{mm}^2
\]
Thus, the area of the regular 15-sided polygon is:
\[
\boxed{810 \, \text{mm}^2}
\]