Chapter 11.5, Problem 29WE

To determine

To Find: The ratio of area of two circles.

Answer to Problem 29WE

  $1:2$

Explanation of Solution

Given: Draw a square and its inscribed and circumscribed circles.

  

Let the side of the square be $s$ .

Then

  $\begin{array}{l}s=2r_1\\ r_1=\frac{s}{2}\end{array}$

Now

Diagonal of the square is.

  $\begin{array}{l}\sqrt{2}s=2r_2\\ r_2=\frac{\sqrt{2}s}{2}\\ r_2=\frac{s}{\sqrt{2}}\end{array}$

Now area of inscribed circle.

  $\begin{array}{l}A=\pi r_1{}^2\\ A=\pi {\left(\frac{s}{2}\right)}^2\\ A_i=\frac{s^2}{4}\pi sq.units\end{array}$

Area of circumscribed circle

  $\begin{array}{l}A=\pi r_2{}^2\\ A=\frac{1}{2}\pi {\left(\frac{s}{\sqrt{2}}\right)}^2\\ A_c=\frac{s^2}{2}\pi sq.units\end{array}$

Ratio of the area of the circle.

  $\begin{array}{l}{A}_c=\frac{s^2}{2}\pi \\ A_i=\frac{s^2}{4}\pi \\ Ratio=\frac{A_i}{A_c}=\frac{\frac{s^2}{4}\pi }{\frac{s^2}{2}\pi }=1:2\end{array}$

Hence, $1:2$