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How to Find the Length of an Arc
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The basic formula for arc length is really
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How to Find the Length of an Arc
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Hi, this is Eric Stone from South Burlington High School in South Burlington, Vermont, for About.com and today, we are going to find the length of an arc.
Calculating the Length of an Arc

So imagine you are riding in your car and you're going around a cloverleaf, and you want to know what distance you go when you start from here, all the way around, from here to here. Well, to figure that out you need to know something called the arc length of a circle. Lets figure out how to figure that out.
Basic Formula for Arc Length

The basic formula for arc length is really straightforward. It is just theta equals s over r, where theta is the angle in radians -- more on that in a second -- and s is the arc length and r is the radius. This is what we are interested in today: we are going to figure out how to find out what the arc length is.

To do that, we need to define some things. First, theta is the measure of an angle, and radians is a different unit of measure than the one you are probably most familiar with, which is degrees. The way to convert from one to the other is simply this: you just have to remember that 180 degrees is equal to pi radians. Once you remember that, the rest is relatively straightforward. All you have to do is take any angle you get -- say, 60 degrees -- and you want to find out how many radians that is, just multiply it by the conversion factor: Sixty degrees times pi radians is 180 degrees. Simplify that out -- 60 divided by 180 is one third -- and you end up with one third pi radians.
Necessary Information to Find the Length of an Arc

So, if I want to find out the arc length I need to know not only the angle in radians, but also the radius. The radius is simply defined, in a circle, as the distance from the center out to the edge. If I want to find how far this car travels as it travels around this circle, from here to here, as long as I know that the radius is 60 feet and as long as I can figure out how many degrees it goes -- and therefore how many radians that is -- I should be able to find the arc length.

Since the formula is theta equals s over r, I can rewrite that to say that the arc length is equal to the radius times theta. The angle is 270 degrees. Why? Because it is three fourths of a 360 degree circle. Look at our picture: from here to here I have gone three quarters of the circle. Therefore, I have gone 270 degrees.

I need to quickly convert my 270 degrees to radians. I have done that by multiplying times pi radians is 180 degrees. Simplify that out and I have three halves pi radians. All I have to do now is finish it up. The arc length is equal to the radius -- which is 60 feet -- times theta, which is three halves pi. That equals 90 pi feet or 283 feet.

And that is how you find the arc length of a segment of a circle!

Thanks for watching! To learn more, visit us on the web at About.com.
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