Question

The area of a sector of a circle of radius $10\text{ cm}$ is $\frac{55}{3}\text{ cm}^2$. The value of central angle is

• A. $\frac{21^\circ}{2}$
• B. $42^\circ$
• C. $105^\circ$
• D. $21^\circ$

Hint:

When solving formulas involving $\pi$, write $\pi$ as $\frac{22}{7}$ to facilitate easier cancellation with numbers like $55$ (since both have common factors of 11).

Answer 1

The Correct Option is D

Step 1: Find the area of the full circle first.
With $r = 10\text{ cm}$ and $\pi = \frac{22}{7}$, the full circle's area is $\pi r^2 = \frac{22}{7} \times 100 = \frac{2200}{7}\text{ cm}^2$.
Step 2: Compare the sector area to the full circle as a plain ratio.
The fraction of the circle this sector covers is $\frac{\text{sector area}}{\text{circle area}} = \frac{55/3}{2200/7} = \frac{55}{3} \times \frac{7}{2200} = \frac{7}{120}$.
Step 3: Convert this fraction of the circle into degrees.
A full circle is $360^\circ$, so the central angle is $\frac{7}{120} \times 360^\circ = 21^\circ$.
\[ \boxed{21^\circ} \]