Find the Radius of a Circle if You Know the Arc Length

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This commodity explains the arc length formula in detail and provides you lot with pace-by-footstep instructions on how to find the arc length. Y'all volition as well learn the equation for sector area.

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

Arc length formula

The length of an arc depends on the radius of a circle and the central bending θ. We know that for the bending 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:

Fifty / θ = C / 2π

As circumference C = 2πr,

L / θ = 2πr / 2π L / θ = r

We observe out the arc length formula when multiplying this equation past θ:

L = r * θ

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

Area of a sector of a circle

We tin find the area of a sector of a circle in a like style. We know that the surface area of the whole circumvolve 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

How to find the length of an arc and sector area: an instance

1. Decide on the radius of your circumvolve. For example, it can be equal to 15 cm. (You lot can also input the diameter into the arc length reckoner instead.)
2. What will be the bending between the ends of the arc? Let's say information technology is equal to 45 degrees, or π/iv.
3. Summate the arc length according to the formula to a higher place: 50 = r * θ = 15 * π/4 = 11.78 cm.
4. Summate the area of a sector: A = r² * θ / two = xv² * π/4 / 2 = 88.36 cm².
5. You can likewise utilize the arc length computer to notice the central angle or the circle's radius. Simply input any two values into the appropriate boxes and watch information technology conducting all calculations for you lot.

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

FAQ

How do yous find arc length without the radius?

To summate arc length without radius, you need the cardinal bending and the sector area:

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

Or the central angle and the chord length:

1. Divide the primal angle in radians past ii and perform the sine role on it.
2. Divide the chord length by double the result of step 1. This adding gives you the radius.
3. Multiply the radius by the central bending to get the arc length.

How do you lot discover arc length using radians?

1. Multiply the central angle in radians by the circumvolve'south radius.
2. That'due south it! The result is simply this multiplication.

How do you calculate arc length without the angle?

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

1. Multiply the area past 2.
2. Then divide the outcome by the radius squared (make sure that the units are the aforementioned) 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 event (in radians).
3. Double the result of the changed sine to get the central angle in radians.
4. One time you have the primal angle in radians, multiply it by the radius to go the arc length.

Does arc length take to be in radians?

Arc length is a measurement of distance, so it cannot exist in radians. The central angle, withal, does not accept to exist in radians. It tin be in whatever unit for angles you like, from degrees to arcsecs. Using radians, even so, 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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Source: https://www.omnicalculator.com/math/arc-length

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