How To Find Arc Length With Circumference And Central Angle

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An arc is whatever portion of the circumference of a circumvolve.^[1] Arc length is the altitude 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 cardinal angle is, you lot tin can hands observe the length of the arc.

1

Prepare the formula for arc length. The formula is ${\text{arc length}}=ii\pi (r)({\frac {\theta }{360}})$ , 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'southward radius into the formula. This information should be given, or you should be able to measure information technology. Make sure you substitute this value for the variable $r$ .
- For case, if the circle'southward radius is x cm, your formula will look like this: ${\text{arc length}}=two\pi (ten)({\frac {\theta }{360}})$ .
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3
Plug the value of the arc'south central angle into the formula. This information should be given, or you should exist able to measure out it. Make sure you are working with degrees, and not radians, when using this formula. Substitute the central angle'south measurement for $\theta$ in the formula.
- For example, if the arc'due south cardinal angle is 135 degrees, your formula volition wait similar this: ${\text{arc length}}=2\pi (ten)({\frac {135}{360}})$ .
four

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

Divide the arc'due south central angle past 360. Since a circle has 360 degrees total, completing this calculation gives you what portion of the entire circle the sector represents. Using this data, you can discover what portion of the circumference the arc length represents.
6

Multiply the 2 numbers together. This will give you lot the length of the arc.

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1

Fix 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'south radius.
2
Plug the length of the circle's radius into the formula. Y'all 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 ten cm, your formula volition look like this: ${\text{arc length}}=\theta (x)$ .
3
Plug the measurement of the arc's central bending into the formula. Y'all should have this data in radians. If y'all know the angle measurement in degrees, y'all cannot utilize this method.
- For example, if the arc's central bending is 2.36 radians, your formula will await like this: ${\text{arc length}}=2.36(ten)$ .
4

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

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Question

How do yous calculate the length of an arc?

Mario Banuelos is an Assistant Professor of Mathematics at California Country Academy, Fresno. With over 8 years of teaching feel, Mario specializes in mathematical biology, optimization, statistical models for genome development, and data science. Mario holds a BA in Mathematics from California State Academy, Fresno, and a Ph.D. in Practical Mathematics from the Academy of California, Merced. Mario has taught at both the high schoolhouse and collegiate levels.

Assistant Professor of Mathematics

Expert Answer
Question

What is an arc length?

It'due south the linear distance measured along the curve of an arc from one stop to the other.
Question

How do I measure the arc central angle in radians?

1 radian = 57.3°, then carve up the primal bending by 57.3° to get radians.
Question

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

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

If the radius of 3 g of a circle makes bending of pi/iv, so what volition be its corresponding arc length?

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

How practice I find the volume of a cylinder?

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

How practise I measure arc length for circular tank if I merely know that the bore of the circle is 18.60 meters?

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

How do I use some other arc to find arc length?

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

How practice I find the arc length of a semicircle if its diameter is nine?

The arc length of a semicircle is half the circumference of the full circumvolve. The circumference of a circumvolve is πd, which in this example is 9π, or 28.26 units. That means the arc length of the semicircle is xiv.xiii units.
Question

If the radius of an arc is 1500 m, how practise I calculate bending?

You would as well need to know the arc length. So use the arc length formula in Method ane above, and piece of work backwards to solve for the angle. Thus the central angle equals the arc length multiplied past 360°, then divided by 2πr.

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If you know the diameter of the circumvolve, yous can withal find the arc length. The formulas for finding arc length apply the circle's radius. Since the radius is half the diameter of a circumvolve, to find the radius, simply separate the diameter by 2.^[4] For case, if the bore of a circumvolve is xiv cm, to find the radius, you would separate 14 by 2:
$14\div two=7$ .
So, the radius of the circle is 7 cm.

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Article Summary X

To find arc length, commencement past dividing the arc's primal angle in degrees by 360. Then, multiply that number past the radius of the circle. Finally, multiply that number by two × pi to discover the arc length. If you want to larn how to summate the arc length in radians, go along reading the commodity!

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