Q. 53 - Calculus

Question

Prove that the area of a sector with central angle $\phi$ in a circle of radius $r$ is given by $A=\frac{1}{2}\phi r^2$.

Step-by-Step Solution

Answer

The area of a sector of a circle is $A=\frac{1}{2}\phi r^2$ which is formulated by using the property of proportionality os central angle and the area.

Step 1. Given Information

The central angle of the sector is $\phi$.

Step 2. Use the proportionality property

• It is known that the area of a circle is $A=\pi r^2$.
• Also, the area of a sector of a circle is proportional to the central angle.
• The central angle of the entire circle is $2\pi$.
• So, the formula of the sector is calculated as shown below:

$\begin{array}{rcl}A\text{'}& =& \frac{\phi }{2\pi }\pi r^2\\ & =& \frac{1}{2}\phi r^2\end{array}$

• So, the area of the sector of a circle is $\frac{1}{2}\phi r^2$.