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°
Example 2 — 60° Sector on a Pizza
Pizza radius = 15 cm, 6 equal slices → each slice angle = 60°
Example 3 — Belt on a Pulley (Radians)
Pulley radius = 0.2 m, belt contact angle = 2.5 rad (~143°)
❓ Frequently Asked Questions
🔗 Related Calculators
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. So 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 of a circle bounded by two radii and the arc between them. It looks like a pizza slice. The area of a sector with central angle θ (radians) and radius r is: A = ½r²θ. For degrees: A = (θ/360) × πr². The sector perimeter = 2r + arc length = 2r + rθ. A semicircle (θ = π) is the most familiar sector.
What is the difference between arc length and chord length?
The arc length is the curved distance along the circle's circumference between two points. The chord is the straight-line distance between the same two points (cutting across the circle). Arc length ≥ chord length, with equality only when θ → 0. Chord length = 2r·sin(θ/2). For a semicircle (θ = 180°), chord = diameter = 2r, and arc length = πr (about 57% longer than the chord).
What is a circular segment?
A circular segment is the region between a chord and the arc it cuts off — the 'bite' taken from a sector when you remove the triangle. Segment area = Sector area − Triangle area = ½r²θ − ½r²sin(θ) = ½r²(θ − sin θ). For a semicircle, the segment area equals the sector area (since the triangle has zero height). Segments appear in engineering: the cross-sectional area of a partially-filled pipe is a circular segment.
How do you convert degrees to radians?
Multiply degrees by π/180. Key conversions: 30° = π/6 ≈ 0.5236 rad; 45° = π/4 ≈ 0.7854 rad; 60° = π/3 ≈ 1.0472 rad; 90° = π/2 ≈ 1.5708 rad; 180° = π ≈ 3.1416 rad; 360° = 2π ≈ 6.2832 rad. Radians are preferred in calculus and physics because derivative formulas for sin and cos are clean only in radians.
Why is the arc length formula s = rθ?
The definition of a radian: 1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. So for 1 radian, arc = 1r; for θ radians, arc = θr. This makes the relationship linear and unit-consistent. It also means the formula is the same regardless of units of length: if r is in metres, s is in metres. This is why radians are the 'natural' angle unit in mathematics.
What is the arc length for a semicircle?
A semicircle has θ = 180° = π radians. Arc length = r × π = πr. For example, a circle with radius 10 cm has a semicircle arc length of 10π ≈ 31.42 cm. The total perimeter of a semicircle (arc + diameter) = πr + 2r = r(π + 2) ≈ 5.14r. This is useful for finding the perimeter of D-shaped or half-round shapes.
What is the arc length formula in terms of degrees?
Substituting θ_rad = θ_deg × π/180 into s = rθ: s = r × θ_deg × π / 180. Equivalently, s = (θ_deg / 360) × 2πr — the fraction of the full circumference corresponding to the angle. Example: 45° arc on a circle of radius 8: s = (45/360) × 2π × 8 = (1/8) × 16π = 2π ≈ 6.28 units.
How is arc length used in real life?
Arc length appears in many practical contexts: (1) Road design — curves in highways are designed using arc length and radius to achieve safe turning speeds. (2) Engineering — the length of a belt or rope wrapped around a pulley is the arc length over the contact angle. (3) Astronomy — angular separations between stars converted to arc lengths on the celestial sphere. (4) Manufacturing — cutting curved parts from sheet metal, or determining wire length needed to wind a coil. (5) Animation — character movement along curved paths uses arc length parameterisation.
Can arc length equal the radius?
Yes — this defines an angle of exactly 1 radian. When the arc length s = r, the central angle θ = 1 radian ≈ 57.296°. This is the geometric definition of the radian: the angle for which the arc length equals the radius. For any angle θ radians, the arc is exactly θ times the radius — which is what makes the radian the most mathematically natural angle unit.
What is the relationship between arc length and circumference?
The circumference is just the arc length for a full circle (θ = 2π radians = 360°). C = 2πr. Any arc length s = (θ/2π) × C — it is the fraction (θ/2π) of the total circumference. This fraction equals θ/360 when θ is in degrees. For example, a 90° arc is exactly ¼ of the circumference: s = (90/360) × 2πr = πr/2.