Arc Length Calculator
Calculate arc length, sector area, chord length, and segment area for any circle radius and central angle.
What Is Arc Length?
The arc length is the distance measured along the curved path of a circle between two points on its circumference. Unlike the straight-line chord connecting those two points, the arc follows the curvature of the circle itself. Arc length is one of the most fundamental measurements in circular geometry, used in engineering, physics, navigation, and design wherever curves appear.
For a circle of radius r and a central angle θ (measured in radians), the arc length is simply:
s = r × θ
This elegant formula shows that arc length scales linearly with both the radius and the angle. This calculator also computes the sector area (the pie-slice region), the chord length (the straight-line shortcut), the sector perimeter (arc + two radii), and the circular segment area (the region between the chord and the arc).
These quantities arise together in many real-world problems: finding the length of a curved road section, calculating material needed to fabricate a curved part, or determining the contact arc of a belt on a pulley.
Arc Length Formulas
All formulas use θ in radians. Convert: θ_rad = θ_deg × π/180.
| Quantity | Formula | Variables |
|---|---|---|
| Arc Length | s = r × θ | r = radius, θ = angle (radians) |
| Sector Area | A = ½ r² θ | Pie-slice area |
| Chord Length | c = 2r sin(θ/2) | Straight line between endpoints |
| Sector Perimeter | P = 2r + rθ | Arc + two radii |
| Segment Area | A = ½ r²(θ − sin θ) | Region between chord and arc |
| Degrees to Radians | θ_rad = θ_deg × π / 180 | — |
For degrees directly: s = (θ_deg / 360) × 2πr = θ_deg × πr / 180
How to Use the Arc Length Calculator
- Enter the radius in any length unit (metres, cm, inches, feet — the result will be in the same unit).
- Select Degrees or Radians for your central angle using the toggle buttons.
- Enter the central angle — must be between 0 and 360° (or 0 to 2π rad).
- Click Calculate to instantly see all six arc and sector measurements with step-by-step working.
Example Calculations
Example 1 — Quarter Circle (90°)
Radius = 10 m, Angle = 90°
1
Convert: 90° × π/180 = π/2 ≈ 1.5708 rad
2
Arc length = 10 × π/2 = 5π ≈ 15.708 m
3
Sector area = ½ × 100 × π/2 = 25π ≈ 78.54 m²
4
Chord = 2 × 10 × sin(45°) = 20 × (√2/2) = 10√2 ≈ 14.142 m
Arc: 15.71 m | Sector area: 78.54 m² | Chord: 14.14 m (¼ of the circle area is 25π m²).
Example 2 — 60° Sector on a Pizza
Pizza radius = 15 cm, 6 equal slices → each slice angle = 60°
1
60° × π/180 = π/3 ≈ 1.0472 rad
2
Arc (crust length) = 15 × π/3 = 5π ≈ 15.71 cm
3
Slice area = ½ × 225 × π/3 = 37.5π ≈ 117.81 cm²
Each pizza slice has a crust of 15.71 cm and an area of 117.81 cm². Six slices: 6 × 117.81 = 706.86 cm² ≈ πr² ✓
Example 3 — Belt on a Pulley (Radians)
Pulley radius = 0.2 m, belt contact angle = 2.5 rad (~143°)
1
θ already in radians: 2.5 rad → 2.5 × 180/π ≈ 143.24°
2
Arc length = 0.2 × 2.5 = 0.5 m of belt in contact
3
Sector area = 0.5 × 0.04 × 2.5 = 0.05 m²
0.5 m of belt wraps the pulley over a 2.5 rad contact angle. This contact length determines friction force capacity (Euler–Eytelwein equation).
❓ Frequently Asked Questions
What is arc length?
Arc length is the distance along the curved boundary of a circle between two points on its circumference. It is the length of the 'bent' path along the circle, not the straight-line chord connecting the two endpoints. For a full circle, the arc length equals the circumference (2πr). For any partial arc with central angle θ (in radians), arc length s = r × θ.
What is the formula for arc length?
Arc length s = r × θ, where r is the radius of the circle and θ is the central angle in radians. If the angle is given in degrees, convert first: θ_rad = θ_deg × π/180. In terms of degrees: s = r × π × θ_deg / 180. Example: radius = 5, angle = 60° → θ_rad = π/3 → s = 5 × π/3 ≈ 5.236 units.
What is a sector of a circle?
A sector is the 'pie slice' region bounded by two radii and the arc between them. Sector area = ½r²θ (θ in radians). For degrees: A = (θ/360) × πr². Sector perimeter = 2r + arc length = 2r + rθ. A semicircle (θ = π radians = 180°) has area = ½πr² and perimeter = πr + 2r = r(π+2).
What is the difference between arc length and chord length?
Arc length is the curved distance along the circle; chord is the straight-line distance between the same two endpoints. Arc length ≥ chord length, equal only when θ → 0. Chord = 2r·sin(θ/2). For a semicircle (θ = 180°), chord = 2r (the diameter) and arc = πr ≈ 3.14r — about 57% longer than the chord.
What is a circular segment?
A circular segment is the region between a chord and the arc — the region you get by removing the triangle from the sector. Segment area = Sector area − Triangle area = ½r²θ − ½r²sin(θ) = ½r²(θ − sin θ). This formula appears in engineering for calculating the cross-sectional area of liquid in a partially-filled horizontal pipe or tank.
How do you convert between degrees and radians?
Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Key values: 30° = π/6 ≈ 0.524 rad; 45° = π/4 ≈ 0.785 rad; 60° = π/3 ≈ 1.047 rad; 90° = π/2 ≈ 1.571 rad; 120° = 2π/3 ≈ 2.094 rad; 180° = π ≈ 3.14159 rad; 270° = 3π/2 ≈ 4.712 rad; 360° = 2π ≈ 6.283 rad.
Why is the arc length formula s = rθ?
By definition of a radian: 1 radian is the angle that subtends an arc equal in length to the radius. So for θ = 1 rad, s = 1 × r = r. For θ radians, s = θ × r. The relationship is linear, and the formula is dimensionally consistent — if r is in metres, s is in metres. This is why radians are the natural unit for circular arc calculations.
What is the arc length of a semicircle?
θ = 180° = π radians. Arc length = r × π = πr. For r = 10 cm: arc = 10π ≈ 31.42 cm. The full perimeter of the semicircular shape (arc + straight diameter) = πr + 2r = r(π + 2) ≈ 5.14r. At r = 10 cm, the perimeter is about 51.4 cm. This is useful for calculating fence lengths around semicircular flower beds or racetrack curves.
How is arc length used in real life?
Arc length appears in: (1) Road engineering — the length of a curved road section for pavement and marking. (2) Belt drives — the contact arc length on a pulley determines friction capacity. (3) Antenna design — dipole and loop antennas are sized by arc length. (4) Animation — movement along curved paths uses arc length parameterisation for constant-speed motion. (5) Manufacturing — cutting curved parts from sheet metal uses arc length to calculate material and cutting time.
Can the arc length be longer than the circumference?
No. The maximum arc length of a circle is the full circumference (2πr), achieved at θ = 360° = 2π radians. This calculator enforces a maximum angle of 360° (or 2π rad). For angles greater than 360°, you would be going around the circle more than once — the "arc" would then be better described as the total path length of multiple revolutions, which would be a multiple of 2πr.
What is the relationship between arc length and the central angle?
They are directly proportional: doubling the central angle doubles the arc length (for the same radius). s = rθ is a linear relationship. This means: a 90° arc is exactly half a 180° arc; a 60° arc is one-sixth of the circumference; an angle of 1 radian (≈57.3°) gives an arc equal to the radius. The ratio s/C = θ/(2π) — the arc is the same fraction of the circumference as the angle is of the full 2π radians.