Question

A regular hexagon has an area of 516 square meters. Find the measure of each side if its apothem is 16 meters long. Submit your answer in decimal form to the nearest hundredth (2 decimal places)



(1 point)
The hexagon has side lengths of
meters.
Question 5
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.(1 point)
The area of the regular 15 sided shape is
mm2
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Answer 1

To find the side length of the regular hexagon, we can use the formula for the area of a regular polygon:

\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]

For a hexagon, if we denote the side length as \( s \), the perimeter \( P \) is given by:

\[
P = 6s
\]

Given that the area is 516 square meters and the apothem is 16 meters, we can substitute into the area formula:

\[
516 = \frac{1}{2} \times (6s) \times 16
\]

Now, simplify the equation:

\[
516 = 48s
\]

Next, solve for \( s \):

\[
s = \frac{516}{48} \approx 10.75
\]

Thus, each side of the hexagon measures approximately **10.75 meters**.

---

Now, for the regular 15-sided polygon, we again use the area formula:

\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\]

Here, the side length \( s \) is 9 millimeters and there are 15 sides, so the perimeter \( P \) is:

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

Now substituting the perimeter and the apothem (12 mm) into the area formula:

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

Calculating this gives:

\[
\text{Area} = \frac{1}{2} \times 1620 = 810 \, \text{mm}^2
\]

Thus, the area of the regular 15-sided shape is **810 mm²**.