Arc Length Calculator - Circle Arc Formula Tool
Calculate the arc length of a circle from its radius and central angle in degrees or radians instantly.
Enter the radius and central angle to compute the arc length. Switch between degrees and radians with one click.
Arc Length Calculator - Circle Arc Formula Tool
Calculate the arc length of a circle from its radius and central angle in degrees or radians instantly.
About the arc length calculator
Arc length is the distance measured along the curved line of a circle between two points on its circumference. It is one of the most fundamental measurements in geometry and trigonometry, appearing everywhere from engineering design to everyday navigation and even the calculation of distances on Earth's surface.
The formula for arc length of a circle is elegantly simple: L = r × θ, where r is the radius of the circle and θ is the central angle measured in radians. When the angle is given in degrees, the formula becomes L = r × (θ × π / 180). This conversion factor π/180 transforms degrees into radians so that the formula remains consistent. A full circle has a central angle of 2π radians (or 360°), so the arc length of a full circle equals 2πr — the familiar circumference formula.
Understanding the relationship between the central angle and arc length is crucial in many fields. In civil engineering, highway curves are designed with specific arc lengths to ensure smooth transitions between straight road sections. In mechanical engineering, gears and pulleys rely on arc length calculations to determine belt lengths and rotational relationships. In astronomy, arc length on the celestial sphere is used to measure angular separations between stars and planets.
The radian is the natural unit for angular measurement in mathematics because it directly relates the angle to the arc length: when θ equals 1 radian, the arc length equals the radius. This is why scientific and engineering calculations overwhelmingly prefer radians over degrees. However, everyday communication and navigation typically use degrees, which is why this calculator supports both.
A quarter circle (90° or π/2 radians) has an arc length of πr/2. A semicircle (180° or π radians) has an arc length of πr. These values arise constantly in practical problems: the curved portion of a running track, the arc of a circular saw blade, the sweep of a radar antenna, or the path traced by a robotic arm.
When precision matters, always verify that the angle value and unit are correctly entered. A common mistake is mixing degrees and radians — for example, entering 90 when you mean 90 degrees but the calculator is set to radians would give an arc length 180/π times larger than intended. This calculator makes the unit selection explicit and visible so that such errors are immediately apparent.
Arc length examples
Three practical scenarios showing how the formula applies to common geometric problems.
| Input | Arc Length | Explanation |
|---|---|---|
| r = 4, θ = 90° | ≈ 6.2832 | Quarter circle with radius 4. L = 4 × (90 × π / 180) = 4 × π/2 ≈ 6.2832 units. |
| r = 10, θ = 180° | ≈ 31.4159 | Semicircle with radius 10. L = 10 × π ≈ 31.4159 units. Half the circumference of the circle. |
| r = 5, θ = 1 radian | 5 | Radius 5 with central angle 1 radian. L = 5 × 1 = 5 units exactly. This shows the elegant property of radians. |
| r = 3, θ = 360° | ≈ 18.8496 | Full circle with radius 3. L = 2π × 3 ≈ 18.8496 units — the complete circumference. |
How to use the arc length calculator
- Enter the radius of the circle in the Radius field. Use any consistent unit such as meters, centimeters, or feet.
- Enter the central angle — the angle at the center of the circle that subtends the arc — in the Central Angle field.
- Select the angle unit: click Degrees if your angle is in degrees (e.g., 90°) or Radians if it is already in radians (e.g., 1.5708).
- Click Calculate Arc Length to see the result. The arc length is returned in the same unit as the radius.
- Click Reset to clear all fields and start a new calculation.
Arc length calculator FAQ
What is the formula for arc length?
For a circle, the arc length is L = r × θ, where r is the radius and θ is the central angle in radians. If the angle is in degrees, use L = r × (θ × π / 180). For a full circle this reduces to the circumference formula L = 2πr.
What is the difference between arc length and chord length?
Arc length is the distance along the curved path of the circle, while chord length is the straight-line distance between the two endpoints of the arc. Arc length is always equal to or greater than chord length. For a semicircle the arc length is πr while the chord length is 2r — a ratio of about 1.57.
How do I convert degrees to radians for this formula?
Multiply the degree value by π/180. For example, 90° × π/180 = π/2 ≈ 1.5708 radians. The calculator does this conversion automatically when you select Degrees as the angle unit, so you can enter the angle in whichever unit is most convenient.
Can I use this calculator for non-circular curves?
This calculator applies the circle arc formula, which is exact for circular arcs. For parabolas, ellipses, or other curves the arc length requires calculus-based integration and a different formula. However, for engineering design where curves are approximated as circular arcs, this calculator gives the correct result.
What units does the arc length come out in?
The arc length is returned in the same unit as the radius. If you enter the radius in meters and the angle in degrees, the arc length is in meters. There is no unit conversion — just be consistent: if the radius is in feet, the result is in feet.
Why is 1 radian special?
One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. This means that when θ = 1 radian, L = r exactly — no multiplication factor needed. This is why radians simplify so many formulas in calculus and physics compared to degrees.