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Question

# A gazebo in the shape of a regular octagon has equal sides of 9 feet and an apothem of 10.9 feet. Find the cost of the gazebo’s floor if the flooring costs \$3 per square foot. Round to the nearest hundred dollars.

Solution

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Given:

\begin{align*} \text{Regular octagon} \\ a=\text{Length apothem}&=10.9\text{ feet} \\ s=\text{Length side}&=9\text{ feet} \\ \text{Cost flooring}&=3\text{ dollars per square foot} \end{align*}

The area of a regular polygon is the product of the length of the apothem and the perimeter, divided by 2.

$A=\dfrac{a\times p}{2}$

The perimeter is the sum of the lengths all sides. An octagon has 8 sides, thus a regular octagon has 8 sides with equal length.

\begin{align*}p=\text{Perimeter}&=\text{Sum lengths of all sides} \\ &=9+9+9+9+9+9+9+9 \text{ feet} \\ &=8\times 9 \text{ feet} \\ &=72 \text{ feet} \end{align*}

We can then determine the area of the regular polygon:

\begin{align*} A&=\dfrac{a\times p}{2} \\ &=\dfrac{10.9\times 72}{2}\text{ square feet} \\ &=\dfrac{784.8}{2}\text{ square feet} \\ &=392.4\text{ square feet} \end{align*}

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