Arc Length Calculator

Find the arc length of a sector by entering the central angle and radius in the calculator below.

How to Calculate Arc Length

Arc length is a measurement of distance along the circumference of a circle or sector between two points. Put another way, an arc is the curved outer edge, or circular portion, of a sector.

A sector is a portion of a circle shaped like a pie slice, composed of two radius line segments and an outer curve of the circle called the arc. A sector also has a central angle, which is the angle between the radii, and a chord, which is the distance between the radii where they meet the arc.

diagram of a sector showing the radius, central angle, chord length, and arc length

Semicircles, quadrants, and slices of pizza or pie are just some examples of sectors.

You can find the length of the arc in a sector using an easy formula.

Arc Length Formula

The following formula defines the arc length of a sector.

arc length (s) = radius (r) × central angle (θ)

Thus, the length of an arc is equal to the radius r of the sector times the central angle θ in radians.

If you know the diameter of the circle, then you can find the radius by dividing it by two.

The central angle needs to be in radians to use this formula. If you have an angle measured in degrees, you can convert it to radians by multiplying the angle in degrees by π divided by 180.

You can also use our degrees to radians converter.

Graphic showing the arc length formula where s is equal to the radius times the central angle in radians

For example, let’s find the arc length of a sector with a radius of 7 and a central angle of 2 radians.

arc length (s) = 7 × 2
arc length (s) = 14

Thus, the arc length s is equal to 14.

Find Arc Length using the Radius and Chord Length

If you know the radius and chord length but you don’t know the central angle, then you need to find the central angle first in order to use the formula above.

You can find the central angle of a sector with the formula:

θ = 2 × sin-1(a / 2r)

The central angle θ in radians is equal to 2 times the inverse sine function of the chord length a divided by 2 times the sector radius r.

Now, you can use the central angle and radius to find the arc length using the formula above.

Find Arc Length using the Central Angle and Chord Length

If you know the central angle and chord length but you don’t know the radius, then you need to find the radius before you can use the arc length formula.

You can find the radius of a sector with the formula:

r = a / 2 × sin(θ ÷ 2)

The radius r of a sector is equal to the chord length a divided by 2 times the sine of the central angle θ divided by 2.

Using the radius and the central angle, you can use the formula above to find the length of the arc.

Find Arc Length using Sector Area and Central Angle

You can also find the length of the arc if the sector area and central angle are known using the formula:

arc length (s) = θ × 2A ÷ θ

The arc length s is equal to the central angle θ in radians, times the square root of 2 times the area A divided by θ.

How to Find Chord Length

A sector is divided into a triangle and outer segment by the chord. The chord, represented as line a in the sector image above, is the line that connects the points where the radii connect with the arc.

The chord length will always be shorter than the arc length, since the chord is the straight-line distance between the two points, while the arc is the curved distance between them.

The chord can be found using the following formula:

chord (a) = 2r × sin(1 / 2θ)

Thus, the length of the chord a is equal to 2 times the radius r times the sine of 1/2 times the central angle θ.

How to Find Sector Area

The area of a sector can be found using the formula:

sector area = r² × θ / 2

The area of a sector is equal to the radius r squared times the central angle θ, divided by 2.

Try our area calculator to calculate the area of a sector and several other shapes.

Major Arc vs. Minor Arc

When two points on a circle divide the circumference into two arcs, the major arc is the larger arc, and the minor arc is the smaller arc.

The major arc is the arc that connects the two points with a central angle greater than 180°, while the minor arc is the arc that connects the two points with a central angle less than 180°.

Graphic showing the major arc and minor arc for a circle

When the two arcs connecting the points measure exactly 180°, the circle is divided into two semicircles. In this case, the arc length is equal to half the circumference.