Question

An arc of a circle having a measure \[{{36}^{\circ }}\] has length 176 m. Find the circumference of the circle.

Answer 1

Hint: To solve the above question, we will first find out what an arc is and the circumference of the circle. Then we will convert \[{{36}^{\circ }}\] into radian by using the formula \[\theta \left( \text{in degrees} \right)=\dfrac{\pi }{180}\times \theta \text{ radians}\text{.}\] After converting the angle into radians, we will apply the formula for the length of the arc which is given by, \[\text{length of the arc}=\dfrac{\theta }{2\pi }\times \left( \text{circumference of the circle} \right)\text{.}\] From here, we will find the circumference of the circle.

Complete step by step solution:
Before, solving the question, we must know what an arc and circumference of the circle are. The circumference of a circle is defined as the linear distance around it. In other words, the perimeter of a circle is its circumference. Now, it is given in the question that when an angle of \[{{36}^{\circ }}\] is subtended on the circumference of the circle, an arc is formed whose length is 176 cm. The first thing we are going to do is convert \[{{36}^{\circ }}\] from degrees to radians. For this, we will use the following conversion
\[\theta \left( \text{in degrees} \right)=\dfrac{\pi }{180}\times \theta \text{ radians}\]
In our case, \[\theta ={{36}^{\circ }},\] so we will get,
\[{{36}^{\circ }}=\dfrac{\pi }{180}\times 36\text{ radians}\]
\[\Rightarrow {{36}^{\circ }}=\dfrac{\pi }{5}\text{ radians}\]
Now, we know that if l is the length of the arc formed when some angle \['\theta '\] is subtended on the circumference of the circle, then l is given by the formula
\[l=\dfrac{\theta \left( \text{in radians} \right)}{2\pi }\times \left( \text{circumference} \right)\]
In our case, l = 176 cm and \[\theta =\dfrac{\pi }{5}\text{ radians}\text{.}\] So, we will get,
\[176cm=\dfrac{\left( \dfrac{\pi }{5} \right)}{2\pi }\times \left( \text{circumference} \right)\]
\[\Rightarrow 176cm=\dfrac{\pi }{10\pi }\times \left( \text{circumference} \right)\]
\[\Rightarrow \text{Circumference}=10\times 176cm\]
\[\Rightarrow \text{Circumference}=1760cm\]
Therefore, the circumference of the given circle is 1760 cm.

Note: The alternate approach to this question is shown below.
We know that when an angle of \[{{360}^{\circ }}\] is subtended on the center, the arc becomes equal to the circumference. Thus, we can say that,
\[\text{Circumference}\propto {{360}^{\circ }}\]
\[\Rightarrow \text{Circumference}=\left( {{360}^{\circ }}k \right)......\left( i \right)\]
where k is constant.
Similarly, when an angle of \[{{36}^{\circ }}\] is subtended, the length of the arc is proportional to the angle \[{{36}^{\circ }}.\] Thus, we have,
\[l\propto {{36}^{\circ }}\]
\[\Rightarrow 176=k\left( {{36}^{\circ }} \right)\]
\[\Rightarrow k=\dfrac{176}{{{36}^{\circ }}}.....\left( ii \right)\]
From (ii), we will put the value of k in (i). Thus, we will get,
\[\Rightarrow \text{Circumference}={{360}^{\circ }}\times \left( \dfrac{176}{{{36}^{\circ }}} \right)\]
\[\Rightarrow \text{ Circumference}=10\times 176cm\]
\[\Rightarrow \text{ Circumference}=1760cm\]