Chapter 1, Problem 59RE

Solution

To find: The area of the circle as function of s.

The area of circle is $\pi \cdot \frac{s^2}{2}$

Given information:

A square of side s inscribed in a circle.

  

The area of square=side $\times$ side

The area of square= $s^2$

In $\Delta ABC\angle B={90}^0$

Apply Pythagoras theorem to find AC

  $\begin{array}{l}A{C}^2=A{B}^2+B{C}^2\\ A{C}^2=s^2+s^2\\ A{C}^2=2s^2\\ AC=\sqrt{2}s\end{array}$

The radius of circle is half of AC

  $\begin{array}{l}r=\frac{1}{2}AC\\ r=\frac{1}{2}\sqrt{2}s\\ r=\frac{s}{\sqrt{2}}\end{array}$

Area of circle= $\pi r^2$

Area of circle= $\pi \cdot \frac{s^2}{2}$

Area of the remaining circle=area of circle-area of square

Area of the remaining circle $=\frac{\pi s^2}{2}-s^2$