Arc length calculator

What is an arc length calculator?

An arc length calculator finds the length of a curved segment along the edge of a circle. The arc is the portion of the circumference that lies between two points on the circle, and its length depends on two things: how far each point is from the center (the radius) and how wide the angle is at the center between them (the central angle).

This calculator works in three directions. If you know the radius and the angle, it returns the arc length. If you know the arc length and one of the other two, it solves for the missing value. You can enter the angle in degrees or radians, and the radius and arc length in any common unit of length.

Key concepts

• Radius (r) — the distance from the center of the circle to a point on its boundary.
• Central angle (θ) — the angle formed at the center of the circle by two radii drawn to the endpoints of the arc.
• Arc length (L) — the distance traveled along the curve from one endpoint of the arc to the other.
• Radian — the natural unit for angles in this formula. One radian is the angle that sweeps out an arc whose length equals the radius. A full circle is $2\pi$ radians, or 360 degrees.

How does the calculator work?

The relationship between arc length, radius, and central angle is linear when the angle is expressed in radians. The calculator converts the angle to radians internally and then applies the formula in whichever direction the user needs.

Formulas

If the angle is in radians:

$$L = r \cdot \theta$$

If the angle is in degrees:

$$L = \frac{\theta}{360} \cdot 2\pi r = \frac{\pi r \theta}{180}$$

Rearranged to solve for the radius:

$$r = \frac{L}{\theta_{\text{rad}}}$$

Rearranged to solve for the angle:

$$\theta_{\text{rad}} = \frac{L}{r}, \qquad \theta_{\text{deg}} = \frac{L}{r} \cdot \frac{180}{\pi}$$

Worked examples

Example 1: arc length from radius and angle

A circle has a radius of 10 cm and you want the length of the arc subtended by a 90° central angle.

$$L = \frac{\pi \cdot 10 \cdot 90}{180} = 5\pi \approx 15.708 \text{ cm}$$

Example 2: arc length from radius and radians

For a radius of 5 m and a central angle of 2 radians:

$$L = 5 \cdot 2 = 10 \text{ m}$$

Example 3: radius from arc length and angle

An arc 15.708 cm long is cut by a 90° angle. The radius is:

$$r = \frac{15.708}{\frac{\pi}{2}} \approx 10 \text{ cm}$$

Example 4: angle from arc length and radius

An arc of 15.708 cm on a circle of radius 10 cm corresponds to:

$$\theta_{\text{rad}} = \frac{15.708}{10} = 1.5708 \text{ rad} = 90°$$

Example 5: full revolution

For radius 1 and angle 360°, the arc length is the full circumference of the circle: $L = 2\pi \cdot 1 \approx 6.2832$.

Practical uses

• Engineering and manufacturing — laying out curved tracks, pipes, belts, or pulleys where a length of curved material must match a known angle.
• Construction and architecture — measuring the curved edges of arches, domes, or roundabout sections.
• Surveying and cartography — computing distances along latitudes or curved boundaries.
• Sewing and pattern making — calculating fabric needed for circular or flared pieces (this is the same calculation that drives the circle sector area calculator).
• Sports — finding the distance an athlete runs around the curved part of a track lane.

Notes

• Radius and angle must both be positive for the result to be meaningful.
• A 0° angle gives an arc length of 0 — the two endpoints coincide.
• When solving for the radius from an arc length and angle, the angle cannot be 0; when solving for the angle, the radius cannot be 0.
• The units of the radius and the arc length match: a radius in metres gives an arc length in metres. Switching the unit selector reconverts the result automatically.