Question

A circle has a sector with area (3/2) pi and central angle of 60°.
What is the area of the circle?
Either enter an exact answer in terms of it or use 3.14 for it and enter your answer as a decimal.

Answer 1

Answer:

$9\pi$, which is approximately $28.26$.

Step-by-step explanation:

Consider two sectors in the same circle. The area of the two sectors is proportional to their central angles. In other words, if the central angle is $\theta_1$ for the first sector in this circle, and $\theta_2$ for the second, then:

$\displaystyle \frac{\text{Area of Sector 1}}{\text{Area of Sector 2}} = \frac{\theta_1}{\theta_2}$.

In this question, think about the whole circle as a sector. The central angle of this "sector" would be $360^\circ$ (a full circle.) Compare the area of this circle to that of the $60^\circ$-sector in this circle:

$\displaystyle \frac{\text{Area of Circle}}{\text{Area of 60^\circ -Sector}} = \frac{360^\circ}{60^\circ} = 6$.

In other words, the area of this circle is six times that of the $60^\circ$-sector in it.

The area of that $60^\circ$-sector is $\displaystyle \frac{3}{2}\pi$. Therefore, the area of this full circle will be $\displaystyle 6 \times \frac{3}{2}\pi = 9\pi \approx 28.26$.