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: Understanding the Question:
The topic of this question is Areas Related to Circles.
A sector of a circle is a region bounded by two radii and an arc of the circle.
If a circle is divided into \(N\) identical (equal) sectors, the area of each individual sector is exactly \(\frac{1}{N}\) of the total area of the circle.
We are given a circle of radius \(r = 7\) cm divided into 16 identical sectors.
We need to calculate the area of each sector.

Step 2: Key Formula or Approach:
1. Calculate the total area of the circle using the standard formula:
\[ \text{Total Area} = \pi r^2 \] 2. Divide this total area by the number of identical sectors (16) to find the area of one sector:
\[ \text{Area of each sector} = \frac{\text{Total Area}}{16} = \frac{\pi r^2}{16} \] We will use the value of \(\pi = \frac{22}{7}\) to perform the calculations.

Step 3: Detailed Explanation:

• Calculate the total area of the circle with radius \(r = 7\) cm:
\[ \text{Total Area} = \pi r^2 \] \[ \text{Total Area} = \frac{22}{7} \times 7 \times 7 \]

• Simplify the expression by cancelling the common factor of 7:
\[ \text{Total Area} = 22 \times 7 = 154 \text{ cm}^2 \]

• Since the circle is divided into 16 identical sectors, calculate the area of each sector:
\[ \text{Area of each sector} = \frac{\text{Total Area}}{16} \] \[ \text{Area of each sector} = \frac{154}{16} \text{ cm}^2 \]

• Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2:
\[ \text{Area of each sector} = \frac{154 \div 2}{16 \div 2} = \frac{77}{8} \text{ cm}^2 \]

Step 4: Final Answer:
The area of each sector is \(\frac{77}{8} \text{ cm}^2\).
Therefore, the correct option is (D).