Question

a 120° sector in a circle has an area of 30 cm². what is the exact area of the circle?
$20\pi$ cm²
$45\pi$ cm²
$10\pi$ cm²
$90\pi$ cm²

Explanation:

Step1: Relate sector to circle area

A sector's area is a fraction of the circle's area. The fraction is $\frac{\text{sector angle}}{360^\circ}$. Let $A$ = area of the circle.
$\text{Sector Area} = A \times \frac{120^\circ}{360^\circ}$

Step2: Substitute given sector area

We know the sector area is $30$ $\text{cm}^2$, so:
$30 = A \times \frac{1}{3}$

Step3: Solve for circle area

Multiply both sides by 3 to isolate $A$:
$A = 30 \times 3 = 90$
Since the full circle area formula is $A = \pi r^2$, the exact area is $90\pi$ $\text{cm}^2$.

Answer:

D. $90\pi \text{ cm}^2$