Question

On a circle of radius $7\mathrm{feet}$, what angle would subtend an arc of length $1\mathrm{feet}$?

Answer 1

Step 1: Find the angle in radians:

Let the radius be $\mathrm{r}=7\mathrm{feet}$

Let the angle made be $\mathrm{\theta }$, and the arc length be $\mathrm{l}=1\mathrm{feet}$.

$$\begin{gathered} \mathrm{length}\mathrm{of}\mathrm{an}\mathrm{arc}=\mathrm{radius}\times \mathrm{angle} \\ \mathrm{l}=\mathrm{r}\times \mathrm{\theta } \\ \mathrm{\theta }=\frac{\mathrm{l}}{\mathrm{r}}\mathrm{radians} \end{gathered}$$

Putting the values accordingly in the above formula,

$$\begin{gathered} \mathrm{\theta }=\frac{1}{7}\mathrm{radians} \\ \mathrm{\theta }=0.1428\mathrm{rad} \end{gathered}$$

Step 2: Find the angle in degrees:

The circumference of a circle is $2\mathrm{\pi r}$

$$\begin{gathered} \mathrm{angle}(\mathrm{in}\mathrm{degrees})=\frac{\mathrm{length}\mathrm{of}\mathrm{arc}\times 360°}{\mathrm{circumference}\mathrm{of}\mathrm{circle}} \\ \mathrm{\theta }=\frac{\mathrm{l}\times 360°}{2\mathrm{\pi r}} \end{gathered}$$

Put the values accordingly in the above formula as shown:

$$\begin{gathered} \Rightarrow \mathrm{\theta }=\frac{1\times 360°}{2\times {\displaystyle \frac{22}{7}}\times 7} \\ \Rightarrow \mathrm{\theta }=\frac{360°}{2\times 22} \\ \Rightarrow \mathrm{\theta }=\frac{360°}{44} \\ \Rightarrow \mathrm{\theta }=\frac{90°}{11} \\ \therefore \mathrm{\theta }=8.1818° \end{gathered}$$

Therefore, the angle that would subtend an arc of length $1\mathrm{feet}$ is $0.1428\mathrm{rad}$ or $8.1818°$.