Chapter 3, Problem 2E

To determine

To calculate: The ratio of area of circle to the area of square.

Answer to Problem 2E

The ratio of area of circle to the area of square is $1.571$.

Explanation of Solution

Given information:

The circle of radius r is circumscribed about a square.

Formula used:

Area of circle whose radius is r is $A=\pi r^2$.

Area of square with side as l units $A=l^2$.

Calculation:

Consider the circle of radius r that is circumscribed about a square.

  

Since, diagonals of square bisect each other at $90°$ . So, side length of square is

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

Area of circle whose radius is r is $A=\pi r^2$.

Area of square with side as l units $A=l^2$.

Therefore,

  $\begin{array}{c}A=l^2\\ =2r^2\end{array}$

Now, ratio of area of circle to area of square is,

  $\begin{array}{c}A=\frac{\pi r^2}{2r^2}\\ =\frac{\pi }{2}\\ =1.571\end{array}$

Thus, the ratio of area of circle to the area of square is $1.571$.