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

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

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}} = (\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}} = (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 (3)/(2)\pi`. Therefore, the area of this full circle will be
`\displaystyle 6 * (3)/(2)\pi = 9\pi \approx 28.26`.