How to Calculate Arc Lengths Without Angles

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The arc length of a circle is the distance along the outside of that circle between two specified points. If you were to walk one-fourth of the way around a large circle and you knew the circle's circumference, the arc length of the section you walked would simply be the circumference of the circle, 2π_r_, divided by four. The straight-line distance across the circle between those points, meanwhile, is called a chord.

If you know the measure of the central angle θ, which is the angle between the lines originating at the center of the circle and connecting to the ends of the arc, you can easily calculate the arc length: L = ( θ/360)×(2π_r_).

The Arc Length With No Angle

Sometimes, however, you are not given θ . But if you know the length of the associated chord c, you can calculate the arc length even without this information, using the following formula:

c = 2_r_ sin (θ/2)

The steps below assume a circle with a radius of 5 meters and a chord of 2 meters.

Solve the Chord Equation for θ

Divide each side by 2_r_ (which equals the diameter of the circle). This gives

c/2_r_ = sin (θ/2)

In this example, (c/2_r_) = (2/[2 x 5]) = 0.20.

Find the Inverse Sine of (θ/2)

Since you now have 0.20 = sin (θ/2), you must find the angle that yields this sine value.

Use your calculator's ARCSIN function, often labeled SIN-1, to do this, or refer too the Rapid Tables calculator (see Resources).

sin-1(0.20) = 11.54 = (θ /2)

23.08 = θ

Solve for the Arc Length

Going back to the equation L = (θ/360) × (2π_r_), input the known values:

L = (23.08/360) × ( 2π_r_) = (0.0641) × (31.42) = 2.014 meters

Note that for relatively short arc lengths, the chord length will be very close to the arc length, as a visual inspection suggests.

References

About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.

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