Arc Length Calculator

The Arc Length Calculator helps you compute the distance along a circular arc, the radius of the circle, or the central angle of a sector. Whether you are a student studying geometry, an engineer designing curved structures, or a DIY enthusiast working on a project, this tool provides instant results with step-by-step solutions and an interactive visual diagram.

What is Arc Length?

Arc length is the distance measured along the curved path of an arc, which is a portion of a circle's circumference. The length depends on two factors: the radius of the circle and the central angle that the arc subtends. A larger radius or larger angle produces a longer arc. The formula s = r x θ (where θ is in radians) gives the arc length. When the angle is in degrees, use s = (π x r x θ) / 180.

Three Calculation Modes

Our calculator offers three modes. Find Arc Length calculates the arc length when you know the radius and central angle. Find Radius determines the radius from a known arc length and angle. Find Angle computes the central angle from the arc length and radius. Each mode shows the complete step-by-step solution so you can follow the mathematical process.

Interactive Diagram

The calculator includes an SVG diagram that updates in real time as you enter values. The purple arc represents the calculated arc, dashed lines show the two radii forming the sector, and the angle indicator displays the central angle. The center dot marks the circle center. This visual feedback helps you understand the relationship between the radius, angle, and arc length.

Real-World Applications

Arc length calculations appear in many fields. Engineers use them for curved roads, railway tracks, and architectural arches. Manufacturers apply them to gear design and belt length calculations. Navigators use arc length for great circle distances on Earth. Physicists analyze rotational motion and pendulum movement using arc length formulas.

Related Circle Geometry Calculators

Arc length is closely related to other circle properties. Explore our Circle Sector Calculator to compute the area and perimeter of a sector, the Circle Arc Calculator for chord length and sagitta, and the Circle Calculator to determine radius, diameter, circumference, and area from any one known value.

Frequently Asked Questions

What is the arc length formula?

The arc length formula is s = r x θ, where s is the arc length, r is the radius, and θ is the central angle in radians. When the angle is in degrees, use s = (π x r x θ) / 180.

How do you convert degrees to radians?

To convert degrees to radians, multiply the angle by π/180. For example, 90 degrees equals π/2 radians (approximately 1.5708). To convert radians to degrees, multiply by 180/π.

What is the difference between arc length and chord length?

Arc length is the distance along the curved path of the arc, measured along the circumference. Chord length is the straight-line distance between the two endpoints. Arc length is always greater than or equal to chord length.

How do you find the radius from arc length and angle?

Use the formula r = s / θ, where s is the arc length and θ is the angle in radians. If the angle is in degrees, first convert to radians by multiplying by π/180, then divide the arc length by this value.

What is a sector of a circle?

A sector is the region of a circle bounded by two radii and the arc between them, like a slice of pizza. The central angle determines the size of the sector. A semicircle (180 degrees) and a quadrant (90 degrees) are special cases.

How is arc length used in real life?

Arc length is used in engineering for curved roads and railways, architecture for domes and arches, manufacturing for gears and pulleys, navigation for calculating distances on Earth, and physics for analyzing rotational motion and pendulums.

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