how to find arc length

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An arc is any portion of the circumference of a circle.^[1] Arc length is the distance from one endpoint of the arc to the other. Finding an arc length requires knowing a bit about the geometry of a circle. Since the arc is a portion of the circumference, if you know what portion of 360 degrees the arc's central angle is, you can easily find the length of the arc.

1

Set up the formula for arc length. The formula is ${\text{arc length}}=2\pi (r)({\frac {\theta }{360}})$ , where $r$ equals the radius of the circle and $\theta$ equals the measurement of the arc's central angle, in degrees.^[2]
2
Plug the length of the circle's radius into the formula. This information should be given, or you should be able to measure it. Make sure you substitute this value for the variable $r$ .
- For example, if the circle's radius is 10 cm, your formula will look like this: ${\text{arc length}}=2\pi (10)({\frac {\theta }{360}})$ .
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3
Plug the value of the arc's central angle into the formula. This information should be given, or you should be able to measure it. Make sure you are working with degrees, and not radians, when using this formula. Substitute the central angle's measurement for $\theta$ in the formula.
- For example, if the arc's central angle is 135 degrees, your formula will look like this: ${\text{arc length}}=2\pi (10)({\frac {135}{360}})$ .
4

Multiply the radius by $2\pi$ 2 π {\displaystyle 2\pi } . If you are not using a calculator, you can use the approximation $\pi =3.14$ for your calculations. Rewrite the formula using this new value, which represents the circle's circumference.^[3]
5

Divide the arc's central angle by 360. Since a circle has 360 degrees total, completing this calculation gives you what portion of the entire circle the sector represents. Using this information, you can find what portion of the circumference the arc length represents.
6

Multiply the two numbers together. This will give you the length of the arc.

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1

Set up the formula for arc length. The formula is ${\text{arc length}}=\theta (r)$ , where $\theta$ equals the measurement of the arc's central angle in radians, and $r$ equals the length of the circle's radius.
2
Plug the length of the circle's radius into the formula. You need to know the length of the radius to use this method. Make sure you substitute the length of the radius for the variable $r$ .
- For example, if the circle's radius is 10 cm, your formula will look like this: ${\text{arc length}}=\theta (10)$ .
3
Plug the measurement of the arc's central angle into the formula. You should have this information in radians. If you know the angle measurement in degrees, you cannot use this method.
- For example, if the arc's central angle is 2.36 radians, your formula will look like this: ${\text{arc length}}=2.36(10)$ .
4

Multiply the radius by the radian measurement. The product will be the length of the arc.

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Question

How do you calculate the length of an arc?

Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels.

Assistant Professor of Mathematics

Expert Answer
Question

What is an arc length?

It's the linear distance measured along the curve of an arc from one end to the other.
Question

How do I measure the arc central angle in radians?

1 radian = 57.3°, so divide the central angle by 57.3° to get radians.
Question

How can I find the central angle when only the radius is given?

It's not possible. You need more information.
Question

If the radius of 3 m of a circle makes angle of pi/4, then what will be its corresponding arc length?

If the arc has a central angle of π/4, and the arc length is the radius multiplied by the central angle, then (3)(π/4) = 3π/4 = 2.36 m.
Question

How do I find the volume of a cylinder?

Multiply the square of the radius by pi and by the height (or length) of the cylinder.
Question

How do I measure arc length for circular tank if I only know that the diameter of the circle is 18.60 meters?

An arc is a portion of the circumference of a circle. If you're asking about the full circumference of the tank, that would be 18.60 multiplied by pi.
Question

How do I use another arc to find arc length?

Assuming the two arcs are part of the same circle, you would have to know the central angles of each arc. The ratio of the two arc lengths would be equal to the ratio of their respective central angles.
Question

How do I find the arc length of a semicircle if its diameter is 9?

The arc length of a semicircle is half the circumference of the full circle. The circumference of a circle is πd, which in this case is 9π, or 28.26 units. That means the arc length of the semicircle is 14.13 units.
Question

If the radius of an arc is 1500 m, how do I calculate angle?

You would also need to know the arc length. Then use the arc length formula in Method 1 above, and work backwards to solve for the angle. Thus the central angle equals the arc length multiplied by 360°, then divided by 2πr.

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If you know the diameter of the circle, you can still find the arc length. The formulas for finding arc length utilize the circle's radius. Since the radius is half the diameter of a circle, to find the radius, simply divide the diameter by 2.^[4] For example, if the diameter of a circle is 14 cm, to find the radius, you would divide 14 by 2:
$14\div 2=7$ .
So, the radius of the circle is 7 cm.

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Article SummaryX

To find arc length, start by dividing the arc's central angle in degrees by 360. Then, multiply that number by the radius of the circle. Finally, multiply that number by 2 × pi to find the arc length. If you want to learn how to calculate the arc length in radians, keep reading the article!

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