Arc Length Calculator

How to Calculate Arc Length

Arc length is the distance along the curved edge of a circle between two points — the curved portion of a sector (a pie-shaped slice of a circle). A sector is bounded by two radii and an arc.

Our calculator supports four different input combinations to find arc length, plus it computes the chord length, sector area, and central angle automatically.

Arc Length Formula

The fundamental formula for arc length is:

s = r × θ

The arc length s equals the radius r times the central angle θ in radians. The angle must be in radians because radians are "unitless" — ensuring both sides have units of distance.

If your angle is in degrees, convert it first:

θ (radians) = θ (degrees) × π / 180

Example: Find the arc length of a sector with radius = 7 and central angle = 90°.

• Convert: 90° × π/180 = π/2 ≈ 1.5708 rad
• Arc length: s = 7 × 1.5708 = 10.9956

Using Radius and Chord Length

If you know the radius and chord but not the angle, find the angle first:

θ = 2 × sin⁻¹(a / 2r)

The central angle θ (in radians) equals 2 times the inverse sine of the chord length a divided by 2 times the radius r. Then use s = r × θ.

Using Central Angle and Chord Length

If you know the angle and chord but not the radius, find the radius first:

r = a / (2 × sin(θ / 2))

The radius equals the chord length a divided by (2 × sine of half the central angle). Then use s = r × θ.

Using Sector Area and Central Angle

If you know the sector area and angle:

r = √(2A / θ)

The radius equals the square root of (2 times the sector area divided by the central angle in radians). Then use s = r × θ.

How to Find Chord Length

The chord is the straight-line distance between the two points where the arc meets the radii. It is always shorter than the arc length (a straight line vs. a curve).

chord (a) = 2r × sin(θ / 2)

The chord length equals 2 times the radius times the sine of half the central angle.

How to Find Sector Area

The area enclosed by the two radii and the arc:

A = r² × θ / 2

The sector area equals the radius squared times the central angle (in radians) divided by 2.

Major Arc vs. Minor Arc

When two points divide a circle into two arcs:

• Minor arc: The shorter arc (central angle < 180°)
• Major arc: The longer arc (central angle > 180°)
• Semicircle: When both arcs are equal (central angle = 180°)

The calculator computes the arc based on the angle you enter. For the major arc, enter the reflex angle (360° minus the minor angle).

Frequently Asked Questions

How do you find arc length using a central angle in degrees?

First convert the angle to radians by multiplying by π/180. Then use the formula s = r × θ with the angle in radians. For example, for a 60° angle with r = 10: θ = 60 × π/180 = π/3 ≈ 1.0472, so s = 10 × 1.0472 = 10.472. You can also use our Angle Converter to convert degrees to radians.

How do you calculate arc length without the radius?

You need the central angle plus either the chord length or the sector area. With the chord: find the radius using r = a / (2 × sin(θ/2)). With the sector area: find the radius using r = √(2A/θ). Then use the standard arc length formula.

Is arc length the same as the angle?

No. The angle measures the rotation between two radii (in degrees or radians), while the arc length measures the actual distance along the curve. Arc length depends on both the angle and the radius — a larger circle with the same angle produces a longer arc.

What is the arc length of 90 degrees?

A 90° arc is exactly one-quarter of the circumference. If the radius is r, then: s = r × π/2 ≈ 1.5708r. For example, with r = 10: s = 10 × π/2 ≈ 15.708.