Question

finding the area of a regular polygon
the regular octagon in the ceiling of this cathedral has
a radius of 10.5 feet and a perimeter of 64 feet.
what is the length of the apothem of the octagon?
round your answer to the nearest tenth of a foot.
feet
using your answer for the length of the apothem, what
is the area of the regular octagon? round your answer
to the nearest tenth of a square foot.
square feet

Explanation:

Step1: Recall the formula for the area of a regular polygon

The formula for the area of a regular polygon is \(A=\frac{1}{2}ap\), where \(a\) is the apothem and \(p\) is the perimeter. Also, we can use the relationship between the radius \(r\), the apothem \(a\) and half - side length \(s/2\). For a regular polygon, if the radius \(r = 10.5\) feet and the perimeter \(p=64\) feet, then the side length \(s=\frac{p}{n}\) (for an octagon \(n = 8\)), \(s=\frac{64}{8}=8\) feet.

Step2: Use the Pythagorean theorem to find the apothem

Let the radius \(r\) be the hypotenuse of a right - triangle, the apothem \(a\) and half - side length \(x=\frac{s}{2}\) be the legs. We know \(x = 4\) feet and \(r=10.5\) feet. By the Pythagorean theorem \(a=\sqrt{r^{2}-x^{2}}\).
Substitute \(r = 10.5\) and \(x = 4\) into the formula: \(a=\sqrt{10.5^{2}-4^{2}}=\sqrt{110.25 - 16}=\sqrt{94.25}\approx9.7\) feet.

Step3: Calculate the area of the regular polygon

We know \(a\approx9.7\) feet and \(p = 64\) feet. Using the formula \(A=\frac{1}{2}ap\).
Substitute \(a = 9.7\) and \(p=64\) into the formula: \(A=\frac{1}{2}\times9.7\times64=9.7\times32 = 310.4\) square feet.

Answer:

The length of the apothem is approximately \(9.7\) feet. The area of the regular octagon is approximately \(310.4\) square feet.