Chapter 13.3, Problem 51SR

To determine

To calculate:

The area of shaded region of given figure.

Answer to Problem 51SR

The area of shaded region is $57.1m^2$ .

Explanation of Solution

Given information:

  

Calculation:

A shaded circle contains an un-shaded square with side length $10m$ . Since, the diagonal of the square is also the diagonal of the square is also the diagonal of the circle, the diagonal of the circle is equal to $10\sqrt{2}m$ because this is the hypotenuse of a right triangle with two side lengths of $10m$ .

The area of the shaded region is equal to the area of the circle minus the area of square.

The square has an area equal to $\left(10m\right)\left(10m\right)=100m^2$ and the circle has an area equal to,

  $\pi {\left(\frac{10\sqrt{2}m}{2}\right)}^2=50\pi m^2$

The area of the shaded region is found by subtracting the area of the square from the area of the circle.

  $\begin{array}{l}\text{Shaded region area}=\text{area of circle-area of square}\\ \text{=}\text{50}\pi m^2-100m^2\\ =57.1m^2\end{array}$

Hence the area of shaded region is $57.1m^2$ .