Chapter 5.6, Problem 27E

To determine

To find: The area of the octagon.

Answer to Problem 27E

  $\text{Area}=70.2u^2$

Explanation of Solution

Given information: Geometry: A regular octagon is inscribed in a circle with radius of 5 feet.

Calculation:

Above is a rough sketch of the original problem.

We are going to be interested in the blue triangle first. The formula for the area of a regular polygon is $A=\frac{1}{2}aP$ where a is the apothem which is the perpendicular distance from the center to one of the sides (the midpoint). The P stands for the perimeter which means we will have to find one of the sides also.

Here is the blue triangle with parts labeled, we are going to split this in half and only look at the right side triangle. When we do the apothem will become one of the new triangle’s sides and the bottom will be ½ of one of the sides of the original octagon.

Now we use our trig ratios to find the apothem and the ½ side.

  $k=\frac{1}{2}bc\sin A$

  $\begin{array}{l}\sin {22.5}^{\circ }=\frac{1/2\text{ side}}{5}\\ 1/2\text{ side}=5\sin {22.5}^{\circ }=1.9\end{array}$

So the length of one side is twice this is 3.5 so the perimeter (since its octagon) is:

  $8\times 3.8=30.4$

Now to find the apothem,

  $\begin{array}{l}\cos {22.5}^{\circ }=\frac{\text{apothem}}{5}\\ \text{apothem}=5\times \cos {22.5}^{\circ }=4.62\end{array}$

Now we can use the formula $A=\frac{1}{2}aP$

  $A=0.5\times 4.62\times 30.4=70.2$

Thus,

  $\text{Area}=70.2u^2$