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Chapter 4: Problem 127

Prove that the area of a circular sector of radius \(r\) with central angle \(\theta\) is \(A=\frac{1}{2} \theta r^{2},\) where \(\theta\) is measured in radians.

Short Answer

Expert verified
The area of a sector formula is derived from the proportion of the sector's central angle to the circle's total angle (which is \(2\pi\) radians), and then this fraction is multiplied by the area of the whole circle. Hence, the formula \(A=\frac{1}{2} \theta r^{2}\) is proven.

Step by step solution

01

Understand the problem

We need to prove the formula for the area of a sector of a circle, \(A=\frac{1}{2} \theta r^{2}\), where \(\theta\) is the central angle in radians and \(r\) is the radius. This formula can be viewed as a proportion of the area of the entire circle (which is \(\pi r^{2}\)), scaled by the fraction of the circle that the sector represents (which is \(\frac{\theta}{2\pi}\) when \(\theta\) is in radians). Therefore, we need to show this relationship.
02

Area of a circle

The entire area of the circle is given by the formula \(A_{circle} = \pi r^{2}\). This represents the area of the circle if it was a full circle with a central angle of \(2\pi\) radians.
03

Establish the ratio of the central angle to the circle's total angle

The ratio of the central angle of the sector to the total angle of the circle is \(\frac{\theta}{2\pi}\). This ratio tells us what fraction of the total circle the sector represents.
04

Area of the sector

Now, to obtain the area of the sector, we simply need to multiply the area of the entire circle by the ratio established in Step 3. \n So, \(A_{sector} = A_{circle} \times \frac{\theta}{2\pi} = \pi r^{2} \times \frac{\theta}{2\pi}\)
05

Simplify

By simplifying the expression obtained in Step 4, we can cancel out the \(\pi\) terms, resulting in \(A_{sector} = \frac{1}{2} \theta r^{2}\). This confirms the formula given initially in the problem.

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