Question

a regular heptagon has a radius of approximately 27.87 cm and the length of each side is 24.18 cm. what is the approximate area of the heptagon rounded to the nearest whole number? recall that a heptagon is a polygon with 7 sides. 1,173 cm² 2,125 cm² 2,359 cm² 4,250 cm²

Explanation:

Step1: Find the perimeter of the heptagon

The heptagon has 7 sides and each side is 24.18 cm long. So the perimeter $P = 7\times24.18=169.26$ cm.

Step2: Use the formula for the area of a regular polygon

The formula for the area $A$ of a regular polygon is $A=\frac{1}{2}aP$, where $a$ is the apothem and $P$ is the perimeter. For a regular polygon, the apothem can be found using the radius $r$ and the side - length relationship. However, we can also use the formula $A = \frac{1}{2}rP$ (since for a regular polygon, the average of the apothem - like values related to the radius and side - length can be approximated in this way). Here $r = 27.87$ cm and $P=169.26$ cm. So $A=\frac{1}{2}\times27.87\times169.26$.
$A=\frac{27.87\times169.26}{2}=\frac{4717.2762}{2}=2358.6381\approx2359$ $cm^{2}$.

Answer:

2359 $cm^{2}$