Chapter 11.3, Problem 48HP

To determine

The sector that makes up 60% of the circle.

Answer to Problem 48HP

The area is calculated by $A=\frac{\pi r^{2}x}{360}$

Explanation of Solution

Given information:

Draw a circle and a sector that makes up 60% of the circle. What is the central angle of the sector?

Formula used:

  $\frac{\text{area of sector}}{\text{Area of circle}}=\frac{1}{2}=\frac{\text{arclength}}{\text{Circumference of circle}}$

Calculation:

Think of how the arc length and the area of a sector are related to the circle as a whole. For instance, half of a circle will have half of the arc length and half of the area of the whole circle. A quarter of a circle will have a quarter of the arc length and a quarter of the area. Express each of these cases mathematically as follows:

Half circle:

  $\frac{\text{area of sector}}{\text{Area of circle}}=\frac{1}{2}=\frac{\text{arclength}}{\text{Circumference of circle}}$

Quarter circle:

  $\frac{\text{area of sector}}{\text{Area of circle}}=\frac{1}{4}=\frac{\text{arclength}}{\text{Circumference of circle}}$

From this the ratio of the area of a sector to the area of the circle should be the same ratio as the arc length divided by the circumference. Next, it is express this mathematically and using known formulas derives the area for a sector. Let A represent the area of the sector.

  

  $\begin{array}{l}\frac{\text{area of sector}}{\text{area of circle}}=\frac{\text{arc length}}{\text{circumference}}\\ \frac{A}{\pi r^2}=\frac{\text{length of arc QRS}}{2\pi r}\\ \frac{A}{\pi r^2}=\frac{\frac{\pi rx}{180}}{2\pi r}\\ A=\frac{\frac{\pi rx}{180}}{2\pi r}\pi r^2\\ A=\frac{\frac{\pi rx}{180}}{2}r\\ A=\frac{\pi rx}{360}r\\ A=\frac{\pi r^{2}x}{360}\end{array}$

Conclusion:

The area is calculated by $A=\frac{\pi r^{2}x}{360}$