How To Find Radius From Arc Length

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An arc is any portion of the circumference of a circle.^[1] Arc length is the distance from ane endpoint of the arc to the other. Finding an arc length requires knowing a bit most the geometry of a circle. Since the arc is a portion of the circumference, if you know what portion of 360 degrees the arc'due south central angle is, y'all can hands find the length of the arc.

1

Gear 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 key angle, in degrees.^[two]
2
Plug the length of the circle's radius into the formula. This data should be given, or you should be able to measure it. Make sure y'all substitute this value for the variable $r$ .
- For case, if the circle's radius is 10 cm, your formula will look like this: ${\text{arc length}}=2\pi (ten)({\frac {\theta }{360}})$ .
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iii
Plug the value of the arc'south primal bending into the formula. This information should be given, or you should be able to mensurate it. Make sure yous are working with degrees, and non radians, when using this formula. Substitute the fundamental angle'south measurement for $\theta$ in the formula.
- For case, if the arc'southward central angle is 135 degrees, your formula volition look similar this: ${\text{arc length}}=2\pi (x)({\frac {135}{360}})$ .
four

Multiply the radius past $two\pi$ . If y'all are not using a calculator, you tin can utilise the approximation $\pi =3.xiv$ for your calculations. Rewrite the formula using this new value, which represents the circle's circumference.^[3]
five

Split up the arc's key angle past 360. Since a circle has 360 degrees total, completing this calculation gives you what portion of the entire circumvolve the sector represents. Using this data, y'all can find what portion of the circumference the arc length represents.
vi

Multiply the two numbers together. This volition give y'all the length of the arc.

1

Ready 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 circumvolve's radius.
2
Plug the length of the circle'due south radius into the formula. You need to know the length of the radius to use this method. Make certain you lot substitute the length of the radius for the variable $r$ .
- For case, if the circumvolve's radius is 10 cm, your formula will expect similar this: ${\text{arc length}}=\theta (10)$ .
three
Plug the measurement of the arc's central angle into the formula. You should accept this data in radians. If you know the angle measurement in degrees, you cannot utilise this method.
- For example, if the arc'due south cardinal angle is 2.36 radians, your formula will wait like this: ${\text{arc length}}=ii.36(10)$ .
4

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

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Question

How practise you calculate the length of an arc?

Mario Banuelos is an Banana Professor of Mathematics at California Land 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.

Banana Professor of Mathematics

Expert Respond
Question

What is an arc length?

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

How do I measure the arc cardinal angle in radians?

ane radian = 57.3°, then dissever the cardinal angle by 57.3° to become radians.
Question

How tin can I discover the key angle when only the radius is given?

It's not possible. Y'all need more data.
Question

If the radius of 3 m of a circumvolve makes angle of pi/4, so what volition 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 (three)(π/4) = 3π/4 = 2.36 m.
Question

How practise I find the volume of a cylinder?

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

How practice I measure out arc length for circular tank if I merely 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 well-nigh the full circumference of the tank, that would exist eighteen.threescore multiplied by pi.
Question

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

Bold the two arcs are function of the same circle, yous would accept to know the central angles of each arc. The ratio of the two arc lengths would be equal to the ratio of their corresponding central angles.
Question

How do I observe 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. So utilise the arc length formula in Method 1 above, and work backwards to solve for the angle. Thus the cardinal angle equals the arc length multiplied by 360°, then divided by 2πr.

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If y'all know the diameter of the circle, you tin 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.^[iv] For example, if the diameter of a circle is 14 cm, to discover the radius, you lot would separate fourteen past 2:
$fourteen\div ii=vii$ .
So, the radius of the circumvolve is vii cm.

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Article Summary X

To discover arc length, start by dividing the arc's primal angle in degrees by 360. Then, multiply that number by the radius of the circle. Finally, multiply that number by two × pi to detect the arc length. If you desire to learn how to calculate the arc length in radians, keep reading the commodity!

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