Question

A circle of diameter $20\text{ cm}$ is equally divided into five sectors. Find the area and perimeter of one of the sectors.

Hint:

Do not forget to add the two radii ($2r = 20\text{ cm}$) to the arc length when calculating the perimeter of a sector.
Failing to do so is a common error in coordinate geometry and mensuration!

Answer 1

Step 1: Understanding the Question:
We are given a circle of diameter $d = 20\text{ cm}$.
This circle is divided into five equal sectors.
We need to calculate both the area and the perimeter of one of these sectors.

Step 2: Key Formula or Approach:
First, find the radius $r$:
\[ r = \frac{d}{2} = 10\text{ cm} \]
- Since the circle is divided into 5 equal parts, the area of one sector is exactly one-fifth of the total area of the circle:
\[ \text{Area of sector} = \frac{1}{5} \pi r^2 \]
- The perimeter of a sector consists of its curved arc length plus the two straight radii bounding it:
\[ \text{Perimeter of sector} = \text{Arc length } (l) + 2r \]
where the arc length is one-fifth of the total circumference:
\[ l = \frac{1}{5} (2\pi r) \]

Step 3: Detailed Explanation:

• Find the radius of the circle:
\[ r = \frac{20}{2} = 10\text{ cm} \]

• Calculate the area of one sector (using $\pi = 3.14$ or leaving in terms of $\pi$):
\[ \text{Area} = \frac{1}{5} \times \pi \times (10)^2 \]
\[ \text{Area} = \frac{100\pi}{5} = 20\pi\text{ cm}^2 \]
Using $\pi = 3.14$:
\[ \text{Area} = 20 \times 3.14 = 62.8\text{ cm}^2 \]

• Calculate the arc length $l$ of one sector:
\[ l = \frac{1}{5} \times 2 \pi r \]
\[ l = \frac{1}{5} \times 2 \times \pi \times 10 = 4\pi\text{ cm} \]
Using $\pi = 3.14$:
\[ l = 4 \times 3.14 = 12.56\text{ cm} \]

• Calculate the perimeter of the sector:
\[ \text{Perimeter} = l + 2r \]
\[ \text{Perimeter} = 4\pi + 2(10) = (4\pi + 20)\text{ cm} \]
Using $\pi = 3.14$:
\[ \text{Perimeter} = 12.56 + 20 = 32.56\text{ cm} \]


Step 4: Final Answer:
The area of one of the sectors is $20\pi\text{ cm}^2$ (or $62.8\text{ cm}^2$) and its perimeter is $(4\pi + 20)\text{ cm}$ (or $32.56\text{ cm}$).