An arc is any curved segment of a circle, and its length is one of the most common quantities in geometry, drafting, engineering, and everyday measurement. Whether you are laying out a curved road, designing a gear, cutting a pizza slice, or solving a high-school trigonometry problem, you often know only two things: the circle's radius and the central angle that marks off the arc. This arc length calculator takes exactly those two numbers and returns the arc length, the same angle in radians, the straight-line chord across the arc, and the area of the sector it encloses.
The core idea is simple: on a unit circle (radius 1), the arc length is exactly the angle measured in radians. Scale that by the real radius, and you have your answer. You never need to work out the full circumference first, although doing so is a useful way to sanity-check the result.
How to Use
- Enter the Radius of the circle. Type the number in the same units you want the answer in — centimeters, inches, meters, or any other linear unit. The arc length and chord come back in those same units, and the sector area in square units.
- Enter the Central Angle value, then pick its unit from the Angle Unit dropdown — Degrees (°) or Radians (rad). The angle is the opening between the two straight sides (radii) of the wedge, measured at the circle's center. A full circle is 360° (2π rad), a semicircle is 180° (π rad), and a quarter circle is 90° (π/2 rad).
- Read the result card from top to bottom. Angle in Radians is your angle converted to radians (the dropdown decides whether a ×π/180 conversion is applied). Arc Length (the highlighted figure) is the main answer. Chord Length is the straight line connecting the two endpoints of the arc, and Sector Area is the pie-slice region bounded by the arc and its two radii.
- Experiment with extreme values to build intuition. Set the angle to 0 and everything collapses to zero; set it to 360° and the arc length becomes the full circumference.
If you already have the angle in radians, choose Radians (rad) from the dropdown and type the value directly — no manual conversion needed. If you have degrees, choose Degrees (°) and the calculator applies the ×π/180 conversion for you.
The Arc Length Formula
Arc length comes directly from the definition of a radian:
s = r × θ
Where:
- s is the arc length.
- r is the radius.
- θ is the central angle measured in radians.
Because most people measure angles in degrees, the calculator first converts degrees to radians:
θ (radians) = angleDegrees × π / 180
For example, 90 degrees is 90 × π / 180 = π/2 radians, about 1.5708 radians. With a radius of 10 cm, the arc length is 10 × 1.5708 = 15.71 cm.
The calculator also derives two closely related quantities:
- Chord length = 2r × sin(θ / 2). This is the straight-line distance between the arc's endpoints. For the same 10 cm radius and 90° angle, the chord is 2 × 10 × sin(45°) = 20 × 0.7071 = 14.14 cm.
- Sector area = ½ × r² × θ. For the same inputs, that is 0.5 × 100 × 1.5708 = 78.54 cm².
A handy rule of thumb: the arc length is always a little longer than the chord, because the curved path between two points is never shorter than the straight one.
Examples
Example 1 — A quarter-circle ramp. A skatepark ramp follows a circular arc with radius 10 meters and a central angle of 90° (a quarter of the full 360° circle, i.e., 25% of the circumference). The calculator converts 90° to π/2 ≈ 1.5708 radians and gives an arc length of 15.71 meters, a chord of 14.14 meters, and a sector area of 78.54 m².
Example 2 — A semicircular arch. A doorway arch is half a circle with radius 5 feet and a 180° central angle. The arc length is 5 × π = 15.71 feet, the chord equals the full diameter, 10 feet, and the sector area is ½ × 25 × π = 39.27 ft².
Example 3 — A small pie slice. A 12-inch pizza (radius 6 inches) cut through the center at a 45° angle gives a slice whose curved edge is 6 × π/4 ≈ 9.42 inches long, with a crust-to-crust chord of about 9.18 inches and a slice area of 56.55 in² — roughly 12.5% of the whole pizza.
Example 4 — A very large circle. For a circle with radius 1,000,000 units and a 180° angle, the arc length reaches 3,141,592.65 units, confirming the engine stays stable at huge magnitudes.
FAQ
How do I calculate arc length without a calculator?
Multiply the radius by the angle in radians. To convert degrees to radians, multiply the degree value by π and divide by 180. So for radius r and angle a in degrees, the arc length is r × a × π / 180. For a 10 cm radius and 90° angle, that is 10 × 90 × π / 180 ≈ 15.71 cm.
What is the difference between arc length and chord length?
Arc length is the distance along the curved edge of the circle; chord length is the straight line between the same two endpoints. The arc is always longer than the chord. The chord formula is 2r × sin(θ/2), where θ is the angle in radians.
How do I convert degrees to radians?
Multiply the degree measure by π and divide by 180. For example, 180° = π radians, 90° = π/2 radians, and 45° = π/4 radians. This calculator performs the conversion automatically and shows the result under "Angle in Radians."
What is a sector area?
A sector is the wedge-shaped region bounded by an arc and its two radii — like a slice of pie. Its area equals ½ × r² × θ, where θ is the central angle in radians. For a 10 cm radius and 90° angle, the sector area is about 78.54 cm².
Can the central angle be larger than 360 degrees?
A single full turn is 360°, so an angle beyond that describes more than one full circle. Mathematically the formula still works — a 720° angle doubles the arc length — but for most practical geometry problems the angle ranges from 0° to 360°. Values above 360° simply wrap around the circle more than once.
What happens when the angle is 0 degrees?
An angle of 0° means the two radii coincide, so the arc, chord, and sector area all collapse to zero. This calculator returns 0 for every output in that case, which is a good sanity check that the formula behaves at its boundary.
Disclaimer
This arc length calculator is provided for informational purposes only and does not constitute engineering, architectural, or professional advice. Results are based on standard Euclidean geometry formulas and the inputs you enter; they are approximate and rounded for display. Always verify critical measurements independently and consult a qualified professional before using these figures in construction, manufacturing, or any high-stakes application.