Question

A circle is divided into 16 identical sectors. If radius of the circle is 7 cm, area of each sector is

• A. \(\frac{77}{4} \text{ cm}^2\)
• B. \(77 \text{ cm}^2\)
• C. \(154 \text{ cm}^2\)
• D. \(\frac{77}{8} \text{ cm}^2\)

Hint:

Alternatively, you can calculate the central angle \(\theta\) of each sector:
Since the circle is divided into 16 equal parts, \(\theta = \frac{360^\circ}{16} = 22.5^\circ\).
The formula for the area of a sector is \(\frac{\theta}{360^\circ} \times \pi r^2\).
Substituting \(\theta\) gives \(\frac{22.5^\circ}{360^\circ} \times \pi r^2 = \frac{1}{16} \times \pi r^2\), which leads to the exact same result but is much more calculation-intensive.
Direct division is always the faster way to solve such problems!

Answer 1

The Correct Option is D

Step 1: Find the arc length of one sector.
Each sector's central angle is \(\theta = \frac{360^\circ}{16} = 22.5^\circ\). Arc length \(l = \frac{\theta}{360^\circ} \times 2\pi r = \frac{22.5}{360} \times 2 \times \frac{22}{7} \times 7 = 2.75\) cm.
Step 2: Use the area formula based on arc length and radius.
A sector's area can also be found from \(\text{Area} = \frac{1}{2} \times r \times l\), which comes directly from treating the sector like a thin triangle with the arc as its curved base.
Step 3: Substitute the values.
\[ \text{Area} = \frac{1}{2} \times 7 \times 2.75 = 9.625 \text{ cm}^2 \]
Step 4: Convert to a fraction and confirm.
\[ 9.625 = \frac{77}{8} \text{ cm}^2 \]
This matches option (D).
\[ \boxed{\frac{77}{8} \text{ cm}^2} \]