Question

A regular dodecagon has an area of 450 square feet. Find each side length if its apothem is 7.5 feet.(1 point)
Each side measures
feet.

Answer 1

To find the side length of a regular dodecagon (12-sided polygon) given the area and apothem, we can use the formula for the area \( A \) of a polygon, which can also be represented in terms of the apothem \( a \) and the perimeter \( P \):

\[
A = \frac{1}{2} \times P \times a
\]

In this case, we know the area \( A = 450 \) square feet and the apothem \( a = 7.5 \) feet. We can rearrange the equation to solve for the perimeter \( P \):

\[
P = \frac{2A}{a}
\]

Substituting the known values into the equation:

\[
P = \frac{2 \times 450}{7.5} = \frac{900}{7.5} = 120 \text{ feet}
\]

Now that we know the perimeter \( P \), we can find the side length \( s \) of the dodecagon. The perimeter of a dodecagon is given by the product of the number of sides and the side length:

\[
P = 12s
\]

So we can set up the equation:

\[
12s = 120
\]

Now, solving for \( s \):

\[
s = \frac{120}{12} = 10 \text{ feet}
\]

Thus, each side measures **10 feet**.