Question

Find in degrees the angle subtended at the center of a circle of diameter 50 cm by an arc of length 11 cm.

Answer 1

Hint:Here first find the radius of circle using relation $2r=d$.Then use the formula $s=r\theta$ to find angle $\theta$, where s is arc length and r is the radius of the given circle and here $\theta$ which will be in radian.Then multiply with $\dfrac{180}{\pi }$ to get a result in degrees.

Complete step-by-step answer:
In the question, we are given a circle of diameter 50 cm and an arc length of 11 cm and we have to find an angle subtended by the arc.
Before proceeding we will first briefly say something about radian.
The radian is an S.I. unit for measuring angles and is the standard unit of angular measure used in areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.
Radian describes the plain angle subtended by a circular arc as the length of arc divided by radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the magnitude in radians of such a subtend angle is equal to the ratio of the arc length to the radius of circle; that is $\theta \ =\ \dfrac{s}{r}$, where $\theta$ is the subtended angle in radians, s is arc length and r is radius .

Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians that is $s=r\theta$.
Now to find the angle we will use formula $s=r\theta$ where s is arc length, r is radius of circle and $\theta$ is degree in radians.
We have the s in 11 cm and diameter is 50 cm, so radius will be 25 cm.
So, the value of $\theta$ is $\ \dfrac{s}{r}$ of $\dfrac{11\ cm}{25\ cm}$ or 0.44 radian.
Now to convert radians into degree we have to multiply with $\dfrac{180}{\pi }$ of $\dfrac{180}{\dfrac{22}{7}}$ or $\dfrac{180\times 7}{22}$.
So, we get, $0.44\times \dfrac{180\times 7}{22}$
Hence, on calculation we get ${{25.2}^{\circ }}$.
The degree subtended by an arc is ${{25.2}^{\circ }}$.

Note: Students generally misunderstand the quantity of $\theta$. Generally, most students have confusion that ‘$\theta$’ in the question is in degree or radian. So, they should clearly know that the value of $\theta$ is in radian.Students should remember to convert from degree to radian one should multiply by $\dfrac{\pi }{180}$ to get the value in radians and to convert from radian to degree one should multiply by $\dfrac{180 }{\pi}$ to get the value in degrees.