Find the area. Round your answer to the nearest hundredth if necessary.
-A regular polygon with 8 sides is circumscribed about a circle of radius 9 inches. Find the area of the polygon.
A) 214.76 square inches
B) 380.29 square inches
C) 275.69 square inches
D) 347.75 square inches

Question

Quizplus · Accepted Answer

The area of a regular polygon that is circumscribed about a circle can be calculated using the formula $Area = \frac{1}{2}P \times a$, where $P$ is the perimeter of the polygon and $a$ is the apothem (the perpendicular distance from the center to a side). For a regular polygon with $n$ sides of length $s$ and radius $r$ of the circumscribed circle, the apothem $a$ can be found using the formula $a = r \times \cos(\frac{\pi}{n})$, and the side length $s$ can be calculated using $s = 2r \times \sin(\frac{\pi}{n})$. For an 8-sided polygon (octagon) circumscribed about a circle of radius 9 inches, first calculate the side length $s$ and then the perimeter $P = n \times s$, and use the apothem $a$ to find the area.Given $n = 8$ and $r = 9$ inches, the side length $s$ is $s = 2 \times 9 \times \sin(\frac{\pi}{8})$, and the apothem $a$ is $a = 9 \times \cos(\frac{\pi}{8})$. Calculating these gives $s \approx 12.7279$ inches and $a \approx 8.09$ inches. The perimeter $P = 8 \times s \approx 101.8232$ inches. Thus, the area is approximately $Area = \frac{1}{2} \times 101.8232 \times 8.09 \approx 214.76$ square inches.

Find the Area