Question

A central angle $\theta$is drawn in a circle of radius $r$, as in the figure below.

(a) The length of the arc subtended by $\theta$is $s=$______.

(b) The area of the sector with a central angle $\theta$is $A=$_______.

Short answer

a. $\mathit{s}\mathbf{=}\mathit{r}\mathit{\theta }$

b.$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{r}}^{\mathbf{2}}\mathit{\theta }$

Step-by-step solution

Part a. Step 1. Given information.

An arc is subtending an angle at the center of the circle.

Part a. Step 2. Concept used.

Here in this question, we will use the relationship between angle and length of the arc of the circle formed by the angle. The relationship is given as below-$\mathrm{length}\left(\mathrm{arc}\right)=\mathrm{radius}\times \mathrm{angle}$

Part a. Step 3. Calculation.

The relationship is-

$\mathrm{length}=\mathrm{radius}\times \mathrm{angle}$

According to the question, length has assumed as $s$, angle as, and $\theta$radius as $r$.

Therefore the relationship can be written as-

$s=r\theta$

Part b. Step 1. Given information.

We have given a circle and a defined angle subtended at the center of the circle.

Part b. Step 2. Concept used.

The area of a circle can be written as area$=\pi r^2$. And for a particular sector of the circle, the area can be determined by using the fraction of the angle covered or subtended. We know the whole area, so we will find the fraction of the angle covered and then multiply it with the complete area of a circle. And the final fraction is our answer.

Part b. Step 3. Calculation.

The area of the circle is: -

area$=\pi r^2$

Angle covered or subtended is $\theta$. Now we have to find the fraction of the angle-

$\mathrm{fraction}=\frac{\theta }{2\pi }$

Now multiply it with the complete area of the circle-

area$=\pi r^2$

$$\begin{gathered} A=\pi r^2\left(\frac{\theta }{2\pi }\right) \\ A=\frac{1}{2}{r}^2\theta \end{gathered}$$