Question

find the perimeter of a sector of a circle with radius 7cm which subtends an angle of 180° at the centre of the circle.

Explanation:

Step1: Recall sector perimeter formula

The perimeter of a sector (\(P\)) is the sum of the length of the arc (\(l\)) and twice the radius (\(2r\)). The formula for the arc length of a sector with radius \(r\) and angle \(\theta\) (in degrees) is \(l=\frac{\theta}{360}\times2\pi r\). So, \(P = \frac{\theta}{360}\times2\pi r+ 2r\).

Step2: Substitute given values

Given \(r = 7\space\text{cm}\) and \(\theta=180^\circ\). First, calculate the arc length:
\(l=\frac{180}{360}\times2\times\frac{22}{7}\times7\) (using \(\pi=\frac{22}{7}\))
Simplify: \(\frac{1}{2}\times2\times22 = 22\space\text{cm}\).

Then, calculate twice the radius: \(2r=2\times7 = 14\space\text{cm}\).

Step3: Sum arc length and twice radius

Perimeter \(P=l + 2r=22 + 14=36\space\text{cm}\).

Answer:

The perimeter of the sector is \(\boldsymbol{36\space\text{cm}}\).