Circle Arc Calculator

What is an Arc of a Circle?

An arc is a portion of a circle's circumference. When you take any two points on a circle, the curved line connecting them along the circle's edge is called an arc. The longer arc between the two points is called the major arc, and the shorter one is the minor arc. The central angle subtended by the arc determines its length — a larger angle creates a longer arc.

The arc length formula is remarkably simple: s = rθ, where s is the arc length, r is the radius of the circle, and θ is the central angle in radians. This formula is one of the most elegant in geometry because it follows directly from the definition of a radian. When the angle equals 1 radian, the arc length equals exactly one radius, making the formula unit-free and exact with no conversion factors needed.

Arc length calculations appear in countless real-world applications. Engineers use them to design curved roads, railway tracks, and roller coasters. Architects calculate arc lengths for arched doorways, domed ceilings, and curved building facades. Manufacturers need arc lengths for cutting curved materials, from pipes to fabric patterns. Even astronomers use arc lengths to measure angular distances in the night sky.

How to Use the Circle Arc Calculator

This calculator solves for any of the three variables in the arc length equation. Select which variable you want to calculate, enter the known values, and the result is displayed instantly with supporting information.

• Solve for Arc Length: Enter the radius and central angle (in degrees or radians) to find the arc length.
• Solve for Radius: Enter the arc length and central angle to determine the circle's radius.
• Solve for Central Angle: Enter the arc length and radius to find the central angle in both degrees and radians.

Arc Length Formula Derivation

The arc length formula is derived from the relationship between the central angle and the full circle. A full circle has a circumference of 2πr and a central angle of 2π radians (360°). The arc length is proportional to the central angle, so s / 2πr = θ / 2π, which simplifies to s = rθ. This proportionality is what makes the formula so simple and intuitive.