Question

An arc 28 cm is cut from a circle of radius 15 cm. Find the angle of the sector formed by this arc.

Answer 1

Answer:

$\displaystyle \frac{28}{15}\; \text{radians}$, which is approximately $107^{\circ}$.

Step-by-step explanation:

In a given circle, the angle of a sector is proportional to the length of the corresponding arc:

$\displaystyle \frac{\text{angle of sector 1 }}{\text{angle of sector 2 }} = \frac{\text{length of arc of sector 1 }}{\text{length of arc of sector 2 }}$.

For the circle in this question, $r = 15\; \rm cm$, and the circumference would be:

$2\, \pi \, r = 30\, \pi\; \rm cm$.

The full circle itself is like a sector with an angle of $2\, \pi$, with the arc length equal to the circumference of the circle.

$\displaystyle \frac{\text{angle of sector}}{\text{angle of full circle}} = \frac{\text{length of arc of sector}}{\text{circumference of circle}}$.

$\displaystyle \frac{\text{angle of sector}}{2\,\pi\; \rm rad} = \frac{28\; \rm cm}{30\, \pi\; \rm cm}$.

Rearrange the equation to find the angle of the sector:

$\begin{aligned} & \text{angle of sector} \\ =\; & (2\,\pi\; \rm rad) \cdot \frac{28\; \rm cm}{30\, \pi\; \rm cm} \\ =\; & \frac{28}{15}\; \rm rad\end{aligned}$.

In other words, the angle of this sector would be $\displaystyle \frac{28}{15}\; \rm rad$. Multiply that measure in radians by $\displaystyle \frac{360^{\circ}}{2\,\pi\; \rm rad}$ to find the value of the angle measured in degrees:

$\begin{aligned} & \frac{28}{15}\; \rm rad \times \frac{360^{\circ}}{2\,\pi\; \rm rad} \approx 107^{\circ}\end{aligned}$.