How to Calculate Arc Length

An arc is any segment on the circumference of a circle. ^{[1] X Research Source} Arc length is the distance from one end of an arc to the other. To find the arc length, you need to have some knowledge of circle geometry. Since the arc is part of the circumference, if you know the angle at the center of the arc it is easy to find out the length of that arc.

Table of Contents

Use the measure of the central angle in degrees

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Create a formula for the arc length. The formula is

arc length

=

2

π

(

r

)

(

θ

360

)

{displaystyle {text{arc length}}=2pi (r)({frac {theta }{360}})}

{displaystyle {text{arc length}}=2pi (r)({frac {theta }{360}})}

, in there

r

{displaystyle r}

r

is the radius of the circle and

θ

{displaystyle theta }

theta

is the measure of the angle at the center of the arc. ^{[2] X Research Source}

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Substitute the value of the length of the circle’s radius into the formula. This information must be given a given assignment, or you can measure it. Remember to replace this value in the variable

r

{displaystyle r}

r

.

For example, if the radius of the circle is 10cm, the formula would look like this: $arc length$ $=$ $2$ $π$ $($ $ten$ $)$ $($ $θ$ $360$ $)$ ${displaystyle {text{arc length}}=2pi (10)({frac {theta }{360}})}$ ${displaystyle {text{arc length}}=2pi (10)({frac {theta }{360}})}$ .

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Substitute the measure of the angle at the center of the arc into the formula. This information must be given a given assignment, or you can measure it. You must use the angle measure in degrees, not radians, when using this formula. Substitute the measure of the central angle in

θ

{displaystyle theta }

theta

in the formula.

For example, if the measure of the angle at the center of the arc is 135 degrees, the formula would look like this: $arc length$ $=$ $2$ $π$ $($ $ten$ $)$ $($ $135$ $360$ $)$ ${displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})}$ ${displaystyle {text{arc length}}=2pi (10)({frac {135}{360}})}$ .

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Multiply the radius by $2$ $π$ ${displaystyle 2pi }$ ${displaystyle 2pi }$ . If you are not using a calculator, an approximate value can be used

π

=

3

,

14

{displaystyle pi =3.14}

{displaystyle pi =3.14}

for calculations. Rewrite the formula with this new value to represent the circumference of the circle. ^{[3] X Research Sources}

For example:
$2$ $π$ $($ $ten$ $)$ $($ $135$ $360$ $)$ ${displaystyle 2pi (10)({frac {135}{360}})}$ ${displaystyle 2pi (10)({frac {135}{360}})}$
$2$ $($ $3$ $,$ $14$ $)$ $($ $ten$ $)$ $($ $135$ $360$ $)$ ${displaystyle 2(3,14)(10)({frac {135}{360}})}$ ${displaystyle 2(3,14)(10)({frac {135}{360}})}$
$($ $62$ $,$ $8$ $)$ $($ $135$ $360$ $)$ ${displaystyle (62.8)({frac {135}{360}})}$ ${displaystyle (62.8)({frac {135}{360}})}$

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Divide the angle at the center of the arc by 360. Since a circle has 360 degrees, this calculation will tell you how much of the arc the arc occupies. With this information, you can find out how much of the arc length takes up the circumference.

For example:
$($ $62$ $,$ $8$ $)$ $($ $135$ $360$ $)$ ${displaystyle (62.8)({frac {135}{360}})}$ ${displaystyle (62.8)({frac {135}{360}})}$
$($ $62$ $,$ $8$ $)$ $($ $0$ $,$ $375$ $)$ ${displaystyle (62.8)(0.375)}$ ${displaystyle (62.8)(0.375)}$

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Multiply two numbers together. You will find the arc length value.

For example:
$($ $62$ $,$ $8$ $)$ $($ $0$ $,$ $375$ $)$ ${displaystyle (62.8)(0.375)}$ ${displaystyle (62.8)(0.375)}$
$23$ $,$ $55$ ${displaystyle 23.55}$ ${displaystyle 23.55}$
So, the arc length of a circle with a radius of 10cm, with a central angle of 135 degrees, is about 23.55 cm.

Use the measure of the central angle in radians

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Create a formula for the arc length. The formula is

arc length

=

θ

(

r

)

{displaystyle {text{arc length}}=theta (r)}

{displaystyle {text{arc length}}=theta (r)}

, in there

θ

{displaystyle theta }

theta

is the measure of the angle at the center of the arc in radians, and

r

{displaystyle r}

r

is the length of the radius of the circle.

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Substitute the value of the length of the circle’s radius into the formula. You need to know the length of the radius to be able to use this method. Remember to substitute the length of the radius into the variable

r

{displaystyle r}

r

.

For example, if the radius of the circle is 10cm, the formula would look like this: $arc length$ $=$ $θ$ $($ $ten$ $)$ ${displaystyle {text{arc length}}=theta (10)}$ ${displaystyle {text{arc length}}=theta (10)}$ .

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Substitute the measure of the angle at the center of the arc into the formula. You must be given this value in radians. If you know the angle measure in degrees then this method cannot be used.

For example, if the measure of the angle at the center of the arc is 2.36 radians, the formula would look like this: $arc length$ $=$ $2$ $,$ $36$ $($ $ten$ $)$ ${displaystyle {text{arc length}}=2,36(10)}$ ${displaystyle {text{arc length}}=2,36(10)}$ .

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Multiply the radius by the radian measure. You will find the arc length value.

For example:
$2$ $,$ $36$ $($ $ten$ $)$ ${displaystyle 2,36(10)}$ ${displaystyle 2,36(10)}$
$=$ $23$ $,$ $6$ ${displaystyle=23,6}$ ${displaystyle=23,6}$
So, the arc length of a circle with a radius of 10cm, with a central angle of 23.6 radians, is about 23.6 cm.

Advice

If you know the diameter of the circle it is still possible to find the arc length. In the formula to calculate the arc length there is the radius of the circle. Since the radius is half the diameter, to find the radius you just need to divide the diameter by 2 ^{[4] X Research Source} . For example, if the diameter of the circle is 14 cm, you would divide 14 by 2 to get the radius:
$14$ $\div$ $2$ $=$ $7$ ${displaystyle 14div 2=7}$ ${displaystyle 14div 2=7}$ .
Therefore, the radius of the circle is 7 cm.

Steps

Use the measure of the central angle in degrees

Use the measure of the central angle in radians

Advice

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