Chapter 5, Problem 71RE

Solution

To calculate: The area inside the circle and outside the hexagon if a regular hexagon is inscribed in a circle.

The area inside the circle and outside the hexagon is $139.1402{\text{cm}}^2$ .

Given information: A regular hexagon is inscribed in a circle. The sides of regular hexagon are $16$ centimeters.

Formula used: The formula to find the area of regular polygon with sides $n$ and side length $x$ is given by $A=\frac{n{x}^2}{4}\cot \left(\frac{180°}{n}\right)$ .

The formula for area of a circle is $A=\pi r^2$ .

Calculation:

A regular hexagon has $6$ sides. So, $n=6$ . It is given that $x=16$ .

Calculate the area of hexagon.

  $\begin{array}{c}{A}_{\text{hexagon}}=\frac{6{\left(16\right)}^2}{4}\cot \left(\frac{180°}{6}\right)\\ =384\cot \left(30°\right)\\ =384\left(1.732\right)\\ =665.1075{\text{cm}}^2\end{array}$

Calculate the radius of the circle.

  $\begin{array}{c}r=\frac{1}{2}\left(x\right)\left(\csc \frac{180°}{n}\right)\\ =\frac{1}{2}\left(16\right)\left(\csc \frac{180°}{6}\right)\\ =8\cdot \csc 30°\end{array}$

Calculate the area of the circle.

  $\begin{array}{c}{A}_{\text{circle}}=\pi \cdot {\left(8\cdot \csc 30°\right)}^2\\ =3.1415{\left(8\cdot 2\right)}^2\\ =3.1415\left(256\right)\\ =804.2477{\text{cm}}^2\end{array}$

To find the area inside the circle and outside the hexagon, subtract the area of hexagon from the area of the circle.

  $\begin{array}{c}\text{Area}=804.2477{\text{cm}}^2-665.1075{\text{cm}}^2\\ =139.1402{\text{cm}}^2\end{array}$

Thus, the area inside the circle and outside the hexagon is $139.1402{\text{cm}}^2$ .