Chapter 12.2, Problem 52HP

To determine

Proof the formula for the length of the arc.

Explanation of Solution

Given:

It is given in the question that the angle will be $\theta$ and distance travelled will be $2r\pi$ .

Concept Used:

In this, use the concept of circumference of a circle which is $2r\pi$ .

Proof:

Since, it is known that an arc is a part of a circle.

The equation of circumference of the circle is $2r\pi$ .

This passes over $2\pi$ radians. The fraction of radians travelled $(\theta )$ over total radians $(2\pi )$ is multiplied by circumference.

This gives $(2r\pi )\times (\frac{\theta }{2\pi })=r\cdot \theta$ when cancelling the variables.

So, the length of the arc will be $s=r\cdot \theta$ .