Question

How do you find the radius of a circle given the length of the arc ?

Answer 1

Hint: The length of arc is directly proportional to the angle covered by the arc, if the angle of arc is $2\pi$ , then the length of the arc is $2\pi r$ where r is the radius of the circle . we already knew that length of arc and angle covered by arc is directly proportional so the proportionality constant is equal to r .

Complete answer:
Let us assume the length of the arc is equal to l.

We know that the length of arc is directly proportional to angle covered by the arc and the proportionality constant is equal to radius of the circle.
We can write the length of the arc is equal to $r\theta$ where r is the radius of the circle and $\theta$ is the angle covered by the arc.
So $l=r\theta$
Further solving we get $r={\displaystyle \frac{l}{\theta }}$
So the radius of the circle is the length of the arc divided by the angle covered by the arc.

Note: In the above question we have seen how the radius of the circle is equal to the length of arc divided by angle covered by arc. Always remember that the angle in this formula is in radian, if the angle given in the question is in degree then convert it into radian and then apply the formula.
The method to convert the degree into radian is to multiply the degree angle with ${\displaystyle \frac{\pi }{180}}$.