Question

Find the area of the sector of a circle when the angle of the sector is $${63}^{\circ }$$ and the diameter of the circle is 20 cm.
A) 35 $$\mathrm{c}{\mathrm{m}}^2$$
B) 45 $$\mathrm{c}{\mathrm{m}}^2$$
C) 55 $$\mathrm{c}{\mathrm{m}}^2$$
D) 65 $$\mathrm{c}{\mathrm{m}}^2$$

Answer 1

Hint:
Here we will use the formula for the area of a sector of a circle. We will substitute the values of $$\theta$$ and $$r$$ in the formula. We will simplify the equation and calculate the area.

Formulas used:
We will use the following formulas:
The diameter of a circle is 2 times its radius:
$$d=2r$$
Area of a sector of a circle with radius $$r$$ and angle $$\theta$$ is given by $$A=\pi r^2{\displaystyle \frac{\theta }{360}}$$.

Complete step by step solution:
First we will draw the circle showing the sector.

We will find the radius of the circle.
Substituting 20 for diameter in $$d=2r$$, we get
$$\begin{array}{l}20=2r\\ \Rightarrow 10=r\end{array}$$
The radius of the circle is 10 cm.
We will now find the area of the given sector.
By substituting $${63}^{\circ }$$ for $$\theta$$, 10 for $$r$$ and $${\displaystyle \frac{22}{7}}$$ for $$\pi$$ in the formula $$A=\pi r^2{\displaystyle \frac{\theta }{360}}$$, we get
$$\Rightarrow A={\displaystyle \frac{22}{7}}\times {\left(10\right)}^2\times {\displaystyle \frac{63}{360}}$$
Simplifying the expression, we get
$$\begin{array}{l}\Rightarrow A={\displaystyle \frac{22\times 100}{40}}\\ \Rightarrow A=55\mathrm{c}{\mathrm{m}}^2\end{array}$$

$\therefore$ Option C is the correct option.

Note:
If we are unable to recall the formula for the area of a sector, we can derive it using the unitary method.
We know that the area of a circle is $$\pi r^2$$ where $$r$$ is its radius. We also know that the angle of a full circle is $${360}^{\circ }$$. We can say that a circle is a sector with an angle of 360 degrees.
The area of a sector with an angle of 360 degrees is $$\pi r^2$$.
The area of a sector with an angle of 1 degree will be $${\displaystyle \frac{1}{{360}^{\circ }}}\cdot \pi r^2$$.
The area of a sector with an angle of $$\theta$$ degrees will be:
$$\begin{array}{l}\Rightarrow A=\theta \times {\displaystyle \frac{\pi r^2}{360}}\\ \Rightarrow A={\displaystyle \frac{\theta }{360}}\times \pi r^2\end{array}$$