Arc Length Calculator - Circle Arc Formula

Formula

L = 2πr(θ/360°)

Arc length is the circumference (2πr) multiplied by the fraction of the circle (θ/360).

Worked Examples

Example 1: Quarter Circle Arc

Problem:A circle has radius 10 cm. Find the length of a 90 degree arc.

Solution:L = 2πr × (θ/360) L = 2π(10) × (90/360) L = 20π × 0.25 L = 5π = 15.708 cm Sanity check: a quarter of the full circumference 2π(10) = 62.832 cm is 15.708 cm. Sector area for the same arc = (θ/360) × πr² = 0.25 × 314.159 = 78.540 cm².

Result:Arc length = 15.708 cm (sector area 78.540 cm²)

Example 2: Sixty Degree Arc

Problem:Find the arc length cut off by a 60 degree central angle on a circle of radius 12 m.

Solution:L = 2πr × (θ/360) L = 2π(12) × (60/360) L = 24π × (1/6) L = 4π = 12.566 m Because 60 degrees is one sixth of a full turn, the arc is one sixth of the circumference 75.398 m. The chord across this arc is shorter: 2r sin(θ/2) = 24 sin(30°) = 12 m exactly, so the arc exceeds the chord by 0.566 m.

Result:Arc length = 12.566 m (chord = 12 m)

Example 3: Working Directly in Radians

Problem:A circle has radius 8 units. Find the arc length for a central angle of 2.5 radians.

Solution:In radians the formula collapses to L = rθ - no 360 anywhere. L = 8 × 2.5 = 20 units Converting to check: 2.5 rad = 2.5 × (180/π) = 143.239 degrees L = 2π(8) × (143.239/360) = 50.265 × 0.39789 = 20.000 units. Same answer. This is exactly why radians exist: one radian is the angle whose arc equals the radius.

Result:Arc length = 20 units (equals rθ)

Frequently Asked Questions

What is Arc Length?

Arc length is the distance along the curved line making up an arc. It is a portion of the circumference of a circle.

What is the formula for Arc Length?

If the angle is in degrees: L = 2πr × (θ/360). If in radians: L = r × θ.

What is a radian?

A radian is a unit of angle measure. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. 2π radians = 360 degrees.

How is it different from chord length?

The arc length measures the curve. The chord length measures the straight-line distance between the two endpoints of the arc.

How do I convert degrees to radians?

Multiply the degrees by π/180. For example, 180° = π radians.

What is the circumference?

The circumference is the arc length of the entire circle (360°). C = 2πr.

Can arc length be infinite?

No, for a finite radius and angle, the length is finite. However, fractal curves can have infinite length in a finite space.

How does radius affect arc length?

Arc length is directly proportional to radius. If you double the radius, the arc length doubles for the same angle.

Background & Theory

Arc Length (L):
The distance along the curved edge of a sector.

Formulas:

Degrees: $L = 2\\pi r \\left(\\frac{\\theta}{360}\\right)$
Radians: $L = r\\theta$

Components:

r (Radius): Distance from center to edge.
θ (Theta): Central angle.
C (Circumference): The full distance around the circle ($2\\pi r$).

The arc length is simply a fraction of the circumference. If the angle is 1/4 of the circle (90°), the arc length is 1/4 of the circumference.

History

The measurement of curves has challenged mathematicians for millennia. Archimedes (c. 250 BC) used the "method of exhaustion" to approximate the circumference of a circle (and thus arc lengths) by inscribing and circumscribing polygons with increasing numbers of sides. This was an early form of calculus. The concept of the radian, which simplifies the arc length formula to L = rθ, was developed much later, appearing in the works of Roger Cotes in 1714.