Question

In a circle, a $60°$ sector has an area of $25\mathrm{\pi }$ $\text{square feet}$. What is the circumference? Leave your answer in terms of $\mathrm{\pi }$ and in simplest radical form.

Answer 1

Step-1: Calculating the radius of circle:

Given in a circle a $60°$ sector has an area of $25\mathrm{\pi }$ square feet.

Area of $60°$ sector $=25\mathrm{\pi }{\text{feet}}^2$

We know that for a circle with radius $r$ the area is $\pi r^2$
$$\begin{gathered} \Rightarrow {\mathrm{\pi r}}^2\times \frac{60°}{360°}=25\mathrm{\pi } \\ \Rightarrow {\mathrm{r}}^2=25\times 6 \\ \Rightarrow \mathrm{r}=\sqrt{150} \\ =12.25\text{feet} \end{gathered}$$

Step-2: Calculating the circumference:

Circumference $=2\mathrm{\pi r}\times \frac{\mathrm{x}}{360°}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{x}\mathbf{=}$sector of circle $\mathbf{]}$
$$\begin{gathered} =2\mathrm{\pi }\times 12.25\times \frac{60°}{360°} \\ =4.08\mathrm{\pi }\text{feet} \end{gathered}$$

Hence, the circumference is $4.08\mathrm{\pi }\text{feet}$.