# Chapter 3 - Radian Measure and the Unit Circle - Section 3.2 Applications of Radian Measure - 3.2 Exercises - Page 113: 58

The new area is twice the size of the original area. Since we solved the question algebraically, we can see that this result holds in general.

#### Work Step by Step

Let $\theta$ be the angle in radians. Let $A$ be the area of the sector. Then the ratio of the angle $\theta$ to $2\pi$ is equal to the ratio of the sector area to the area of the whole circle. $\frac{\theta}{2\pi} = \frac{A}{\pi ~r^2}$ $A = \frac{\theta ~r^2}{2}$ Let the original angle be $\theta_1$ and let the original radius be $r_1$: $A_1 = \frac{\theta_1 ~r_1^2}{2}$ We can find an expression for the new area: $A_2 = \frac{\theta_2 ~r_2^2}{2}$ $A_2 = \frac{(\theta_1/2) ~(2r_1)^2}{2}$ $A_2 = 2\times \frac{\theta_1 ~r_1^2}{2}$ $A_2 = 2~A_1$ The new area is twice the size of the original area. Since we solved the question algebraically, we can see that this result holds in general.

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