Chapter 10.4, Problem 47HP

To determine

To explain:

The ratio of the area of circle to the area of the square inscribed in a circle

Answer to Problem 47HP

The ratio of the area of circle to the area of the square inscribed in a circle is $\frac{\pi }{2}$ .

Explanation of Solution

Here, given that:

The square is in inscribed in a circle of radius $\left(r\right)$ with side $\left(s\right)$

as:

  

Use the Pythagoras theorem as:

  $\begin{array}{l}{s}^2=r^2+r^2\\ s^2=2r^2\end{array}$

  $s=r\sqrt{2}$  ... (1)

Now, the area of the square of side $\left(s\right)$ is given by the following expression:

  $A_s=s^2$  ... (2)

The area of the circle of radius $\left(r\right)$ is given by the following expression:

  $A_c=\pi r^2$  ... (3)

Now, the ratio of the area of the circle to area of square is calculated as:

  $\frac{A_c}{A_s}=\frac{\pi r^2}{s^2}$

From eq. (1):

  $\begin{array}{l}\frac{A_c}{A_s}=\frac{\pi r^2}{{\left(r\sqrt{2}\right)}^2}\\ \frac{A_c}{A_s}=\frac{\pi r^2}{2r^2}\\ \frac{A_c}{A_s}=\frac{\pi }{2}\end{array}$