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Chapter 4: Problem 78

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 circular sector of radius \(r\) with central angle \(\theta\) is \(\frac{1}{2} \theta r^{2}\), where \(\theta\) is measured in radians.

Step by step solution

01

Recall the Definitions

Recall essential definitions:- The area of a circle is given by \(A_{circle} = \pi r^{2}\) where \(r\) is the radius of the circle.- The circumference of a circle is given by \(C = 2\pi r\) where \(r\) is the radius of the circle.- The extent of the circle cut out by an angle of 1 radian is equal to the radius of the circle. Therefore, \(1\) radian cuts out a circumference \(C_{radian} = r\).- The ratio of the area of a sector to the area of the circle is the same as the ratio of the circumference cut by \(\theta\) radians to the total circumference of the circle.
02

Calculate Ratio of Circumference

Calculate the ratio of the circumference cut out by \(\theta\) radians to the total circumference of the circle. Since \(1\) radian cuts out a length of \(r\), while the total circumference is \(2\pi r\), the ratio \(R\) is equal to \(\frac{\theta r}{2\pi r} = \frac{\theta}{2\pi}\).
03

Calculate Area of Sector

Calculate the area of the sector of radius \(r\) and central angle \(\theta\). Since the ratio of areas is equal to the ratio of corresponding lengths, the area of the sector \(A_{sector}\) is equal to \(R \times A_{circle}\), or \(\frac{\theta}{2\pi} \times \pi r^{2} = \frac{1}{2} \theta r^{2}\).

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