# Arc lenght

## Homework Statement

I dont understand how they came up with this formula.... can some one show me the proof how they derived this formula or can someone send me a link and I will just read it. Thank you.

## The Attempt at a Solution

Well, this formula is the DEFINITION of the unit 'radians'. The theta that you mentioned in this formula is measured in radians.

The measurement of an angle in 'radians' is defined as the length of the arc that it subtends divided by the radius.

Just by definition, plug in theta with respect to it's radians and compare it to the cirumference formula (there is a a lot of similiarty)
$$C = 2\pi r$$
Now if you use the formula
$$S_{arc} = \theta_{rad} r$$
and plug in a degree $$\theta_{rad}$$ with respect to radians you end up with some result
$$N\pi r$$
where N is a rational number.

Ok... but what is the mathamatical proof.

Ok... but what is the mathamatical proof.

There is no mathematical proof! That's how the quantities have been defined!

Redbelly98
Staff Emeritus
Homework Helper
Perhaps what is really wanted is a proof that arc length is proportional to the product of the angle and the radius.

Perhaps what is really wanted is a proof that arc length is proportional to the product of the angle and the radius.

That would be nice..

Redbelly98
Staff Emeritus