How to Calculate the Length of an Arc

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There are many ways to find the length of an arc, and the calculation needed depends on what information is given at the start the problem. The radius is usually the defining starting point, but there are examples of all types of formulas that you can use to solve arc length trig problems.

    Define your terms and give set variable titles so we can understand the formulas quickly. Diameter is the distance across the circle. Its variable is d. Circumference is the distance around the circle; variable c. Area is the space inside the circle; variable A. Radius is half way across the circle or half the diameter; variable r. Theta is the angle given inside the circle, either in radians or in degrees; variable ?. The variable for length of an arc will be s.

    Skip this step, if the radius is given. Below are all the ways to find the radius using other information about the arc. r= d/2 r=c/2? r=?(A/?) So if we have the diameter, the circumference, or the area of the circle, we can find the radius.

    Calculate the length of the arc. Now that we know the radius, we can easily find the length of the arc. If the angle of the arc is given in radians we use the formula: s= ?r If the angle of the arc is given in degrees we use the formula: s= (?/360) x 2 ?r

    Try Example 1. Let’s say our circle has a circumference of 6 and an angle of ?/2. First remember that r= c/2?. Plug 2 in for c so r=2/2?. r= .318 Length would be s = ?r ?= ?/2 r = .318 s= ?/2 x .318 s=.49 Our length of the arc is .49.

    Try Example 2. Now we have a different circle with an Area of 25 and an angle of 80?. To find the radian we use the formula r=?(A/ ?). 25(area) /3.14(pi) = 7.96 ?7.96 =2.82
    r=2.82 Now we use the equation s= (?/360) x 2 ?r s=(80/360) x 2(3.14)(2.82) s=.22 x 17.71 s = 3.94
    Our length is 3.94.

About the Author

Suzanne Hodgson is a graduate of Penn State University, receiving Bachelor of Arts degrees in journalism and integrative art. She works at a marketing firm and has previously served as the photographer and journalist for the "Kennebunk Post," a weekly paper in southern Maine.

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