What is the angle subtended by the arc at the centre if the length of the arc of a circle is equal to the radius of the circle?

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:

As circumference C = 2πr,

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

The formula for the area of a sector is:

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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An arc of a circle is a "portion" of the circumference of the circle. The length of an arc is simply the length of its "portion" of the circumference. The circumference itself can be considered a full circle arc length.

Arc Measure: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc.

Arc Length: In a circle, the length of an arc is a portion of the circumference.   The letter "s" is used to represent arc length.

Consider the following proportion:

If we solve the proportion for arc length, and replace "arc measure" with its equivalent "central angle", we can establish the formula:

Notice that arc length is a fractional part of the circumference. For example, an arc measure of 60º is one-sixth of the circle (360º), so the length of that arc will be one-sixth of the circumference of the circle.

In circle O, the radius is 8 inches and minor arc

is intercepted by a central angle of 110 degrees. Find the length of minor arc

to the nearest integer.
    

As you progress in your study of mathematics and angles, you will see more references made to the term "radians" instead of "degrees". So, what is a "radian" ?

The radian measure, θ, of a central angle is defined as the ratio of the length of the arc the angle subtends, s, divided by the radius of the circle, r.      which gives arc length, s:     s = θr	subtend = "to be opposite to"
One radian is the central angle that subtends an arc length of one radius (s = r). Since all circles are similar, one radian is the same value for all circles.

Relationship between Degrees and Radians: In a circle, the arc measure of the entire circle is 360º and the arc length of the entire circle is represented by the formula for circumference of the circle: . Substituting C into the formula s = θr shows: C = θr         2πr = θr         2π = θ The arc measure of the central angle of an entire circle is 360º and the radian measure of the central angle of an entire circle = 2π. 360º (degrees) = 2π (radians)

360º = 2π (divide both sides by 2) 180º = π

360º = 2π (divide both sides by 4)

360º = 2π (divide both sides by 360)

360º = 2π (divide both sides by 2π)

To change from degrees to radians, multiply degrees by

To change from radians to degrees, multiply radians by

1. Convert 60º to radians.	2. Convert 135º to radians.
3. Convert to degrees.	4. Convert to degrees.
5. Find the length of an arc subtended by an angle of radians in a circle of radius 20 centimeters.

The diagram at the right shows two circles with the same center (concentric circles). It has already been shown that concentric circles are similar under a dilation transformation. The ratio of similitude of the smaller circle to the larger circle is:

The same dilation that mapped the smaller circle onto the larger circle will also map the slice (sector) of the smaller circle with an arc length of s1 onto the slice (sector) of the larger circle with an arc length of s2. When the radius gets dilated by a scale factor, the arc length is also dilated by that same scale factor.

As long as the central angles are the same, the slices (sectors) will be similar.

Since corresponding parts of similar figures are in proportion,

In relation to the two arc length formulas seen on this page, both show that arc length, s, is expressed as "some value" times the radius, r. The arc length is proportional to the radius.

When θ is in degrees:

When θ is in radians:

Setting r = 1 shows the constant of proportionality.