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: Understanding the Question:
We are given the area of a sector of a circle as $\frac{55}{3}\text{ cm}^2$.
The radius ($r$) of this circle is $10\text{ cm}$.
We need to determine the value of the central angle ($\theta$) subtended by this sector in degrees.

Step 2: Key Formula or Approach:
The formula for the area of a sector of a circle is:
\[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \]
We will substitute the given values into this formula and solve for $\theta$.

Step 3: Detailed Explanation:

• Identify the given parameters:
\[ \text{Area of sector} = \frac{55}{3}\text{ cm}^2 \]
\[ \text{Radius, } r = 10\text{ cm} \]
We will use $\pi = \frac{22}{7}$ for calculations.

• Substitute these values into the sector area formula:
\[ \frac{55}{3} = \frac{\theta}{360^\circ} \times \frac{22}{7} \times (10)^2 \]
\[ \frac{55}{3} = \frac{\theta}{360^\circ} \times \frac{2200}{7} \]

• Isolate the variable $\theta$:
\[ \theta = \frac{55}{3} \times \frac{360^\circ \times 7}{2200} \]

• Simplify the fraction step-by-step:
- Simplify $\frac{360^\circ}{3}$:
\[ \theta = 55 \times \frac{120^\circ \times 7}{2200} \]
- Divide 2200 by 55:
Since $55 \times 4 = 220$, we have $\frac{2200}{55} = 40$.
\[ \theta = \frac{120^\circ \times 7}{40} \]
- Simplify $\frac{120^\circ}{40}$:
\[ \theta = 3 \times 7 = 21^\circ \]


Step 4: Final Answer:
The value of the central angle is $21^\circ$.