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: Find the central angle of one sector instead of using the fraction 1/5 directly.
Radius $r = \frac{20}{2} = 10\text{ cm}$. Since the circle is split into 5 equal sectors, each sector has a central angle of $\frac{360^\circ}{5} = 72^\circ$.
Step 2: Find the sector area using the angle formula. \[ \text{Area} = \frac{72^\circ}{360^\circ} \times \pi r^2 = \frac{1}{5} \times 3.14 \times 100 = 62.8\text{ cm}^2 \]
Step 3: Find the arc length using the angle formula. \[ l = \frac{72^\circ}{360^\circ} \times 2\pi r = \frac{1}{5} \times 2 \times 3.14 \times 10 = 12.56\text{ cm} \]
Step 4: Add the two bounding radii to get the perimeter. \[ \text{Perimeter} = l + 2r = 12.56 + 20 = 32.56\text{ cm} \]
\[ \boxed{\text{Area} = 62.8\text{ cm}^2, \ \text{Perimeter} = 32.56\text{ cm}} \]