Question

A regular dodecagon has an area of 450 square feet. Find each side length if its apothem is 7.5 feet.

Answer 1

To find the side length of a regular dodecagon (12-sided polygon) given its area and the apothem, we can use the formula for the area of a regular polygon:

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

Where:
- \(A\) is the area of the polygon,
- \(P\) is the perimeter of the polygon,
- \(a\) is the apothem.

We are given:
- \(A = 450\) square feet,
- \(a = 7.5\) feet.

First, we can rearrange the formula to solve for the perimeter \(P\):

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

Substituting the values:

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

Next, we compute the length of each side of the dodecagon. The perimeter of a regular dodecagon with side length \(s\) is given by:

\[
P = 12s
\]

Setting this equal to the calculated perimeter:

\[
12s = 120
\]

Now we can solve for \(s\):

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

Thus, the length of each side of the regular dodecagon is:

\[
\boxed{10 \text{ feet}}
\]