Arc Length of a Circle: Learn the Proof!

[Miike012]

Homework Statement

Today we went over finding the arc length s of a circle with a given radian and radius...
Thus s = radian*radius...

Thats easy to remember but I think it will be more memorable for the long run if I knew the proof and understood it... can some one please post a website where I can read the proof... or if some one could explain that would be nice to .
Thank you.

Homework Equations

The Attempt at a Solution

 

[Mark44]

The arc length of the entire circle (its circumference) of radius r is $2 \pi r$. IOW, the arc length associated with an angle of $2 \pi$ is $2 \pi r$. Since the arc length is proportional to the angle between the two rays that subtend the arc, the arc length associated with an angle $\theta$ is $\theta r$.

So s = radius * (angle measure in radians).

 

[Miike012]

But this formula only works if I am presented with radians correct? So if I am given deg. I will have to convert into rad right?

 

[Char. Limit]

But this formula only works if I am presented with radians correct? So if I am given deg. I will have to convert into rad right?

That's correct. Alternately, you could use this formula:

$$s=\frac{180}{\pi} \theta r$$

Where $\theta$ is measured in degrees.

 

[Miike012]

Say 64 is the deg. w/ radius of 1
Then your saying I can multiply 180/pi*64*1
= 3666... that seems to big to be an arc length of radius 1

 

[eumyang]

That's because the formula is wrong. It should be
$$s=\frac{\pi}{180} \theta r$$

 

[Miike012]

Thank you.

 

[Char. Limit]

That's because the formula is wrong. It should be
$$s=\frac{\pi}{180} \theta r$$

Oh, my bad... I got the conversion wrong, I guess.

Sorry, Miike!

 

[Miike012]

Oh, my bad... I got the conversion wrong, I guess.

Sorry, Miike!

Its cool no big deal