How to find the radius of a sector when given the arc length and angle

Updated April 24, 2017

By Chance E. Gartneer

The sector of a circle is a partition of that circle. A sector extends from the center, or origin, of the circle to its circumference and encompasses the area of any given angle that also originates from the center of the circle. A sector is best thought of as a piece of pie, and the bigger the angle of the sector, the bigger slice of pie. Each side of the segment is a radius of the circle. You can find the radius of both the sector and the circle by using the sector's angle and area.

    Double the area of the segment. For example, if the segment area is 24 cm^2, then doubling it results in 48 cm^2.

    Multiply the sector's angle by π, which is a numerical constant that begins 3.14, then divide that number by 180. For the example, the sector's angle is 60 degrees. Multiplying 60 by π results in 188.496, and dividing that number by 180 results in 1.0472.

    Divide the area doubled by the number obtained in the previous step. For the example, 48 divided by 1.0472 results in 45.837.

    Find the square root of that number. For the example, the square root of 45.837 is 6.77. The radius of this segment is 6.77 cm.

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

In case you're new to circles, calculating the length and area of sectors could be a little advanced, and you need to start with simpler tools, such as circle length and circumference and area of a circle calculators.

The length of an arc depends on the radius of a circle and the central angle θ. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:

L / θ = C / 2π

As circumference C = 2πr,

L / θ = 2πr / 2π L / θ = r

We find out the arc length formula when multiplying this equation by θ:

L = r * θ

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,

A / θ = πr² / 2π A / θ = r² / 2

The formula for the area of a sector is:

A = r² * θ / 2

  1. Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)
  2. What will be the angle between the ends of the arc? Let's say it is equal to 45 degrees, or π/4.
  3. Calculate the arc length according to the formula above: L = r * θ = 15 * π/4 = 11.78 cm.
  4. Calculate the area of a sector: A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm².
  5. You can also use the arc length calculator to find the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

Make sure to check out the equation of a circle calculator, too!

To calculate arc length without radius, you need the central angle and the sector area:

  1. Multiply the area by 2 and divide the result by the central angle in radians.
  2. Find the square root of this division.
  3. Multiply this root by the central angle again to get the arc length.
  4. The units will be the square root of the sector area units.

Or the central angle and the chord length:

  1. Divide the central angle in radians by 2 and perform the sine function on it.
  2. Divide the chord length by double the result of step 1. This calculation gives you the radius.
  3. Multiply the radius by the central angle to get the arc length.

  1. Multiply the central angle in radians by the circle’s radius.
  2. That’s it! The result is simply this multiplication.

To calculate arc length without the angle, you need the radius and the sector area:

  1. Multiply the area by 2.
  2. Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians.

Or you can use the radius and chord length:

  1. Divide the chord length by double the radius.
  2. Find the inverse sine of the result (in radians).
  3. Double the result of the inverse sine to get the central angle in radians.
  4. Once you have the central angle in radians, multiply it by the radius to get the arc length.

Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

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This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level.

I know how you find out the Area of a Sector and the Arc Length but I'm not sure how to find out the radius of a circle?

I understand that there are formulas but I find them quite confusing. Please can you explain the answer to this for me.

Thanks.

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