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

User Morocklo
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4 votes

Answer:


\displaystyle (28)/(15)\; \text{radians}, which is approximately
107^(\circ).

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} = (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 (28\; \rm cm)/(30\, \pi\; \rm cm) \\ =\; & (28)/(15)\; \rm rad\end{aligned}.

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


\begin{aligned} & (28)/(15)\; \rm rad * (360^(\circ))/(2\,\pi\; \rm rad) \approx 107^(\circ)\end{aligned}.

User ThatMSG
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