The Arc Length Calculator computes the length of a circular arc from the radius and central angle in degrees or radians. Used in geometry, engineering design, road layout, manufacturing, and any application requiring the curved distance along a circular path.
15.708
units
1.570796
rad
0.2500
15.708
units
1.570796
rad
0.2500
The straight-line distance across a curve is not the distance along it. Whether you are designing a curved road, cutting a circular gasket, programming a CNC machine tool path, or solving a geometry problem, you need the arc length — the actual distance traveled along the curve. The calculator for arc length computes this distance from the circle's radius and the central angle that defines the arc.
Arc length s is directly proportional to both the radius r and the central angle θ:
s = r × θ (when θ is in radians)
s = r × θ × (π/180) (when θ is in degrees)
s = 2πr × (θ°/360°) (fraction of full circumference)
All three formulas are equivalent — the radian formula is the simplest and most fundamental. For a circle of radius 5 m with a central angle of 72°: s = 5 × 72 × π/180 = 5 × 1.2566 = 6.283 m. This is exactly 1/5 of the full circumference (2π × 5 = 31.416 m), because 72° = 1/5 of 360° — a useful sanity check. Use this online calculator for any radius and angle combination. The arc length of curve calculator handles the general case for non-circular curves using calculus integration.
Arc length calculations are embedded in engineering design across multiple disciplines:
The ellipse calculator and plane geometry calculators provide complementary geometric calculations.
The circular sector — the "pie slice" region bounded by two radii and an arc — has area directly related to arc length:
Sector area A = ½ × r × s = ½ × r² × θ (θ in radians)
For the same 5 m radius, 72° arc: A = ½ × 5² × (72 × π/180) = ½ × 25 × 1.2566 = 15.71 m². This equals 1/5 of the full circle area (π × 25 = 78.54 m²), consistent with 72° = 1/5 of 360°. The sector area formula is used in surveying (pie-shaped land parcels), fluid dynamics (turbine blade swept area), and optics (solid angle calculations).
On a sphere, the arc length along a great circle (the shortest path between two points on the surface) is analogous to planar arc length: s = R × θ, where R is the sphere's radius and θ is the central angle in radians between the two points. For Earth (R ≈ 6,371 km), 1 radian of arc = 6,371 km; 1° of arc ≈ 111.1 km; 1 arcminute ≈ 1.852 km = exactly 1 nautical mile by definition. This spherical arc length formula underpins all great-circle navigation calculations used in aviation and maritime route planning.
The Arc Length Calculator uses the fundamental arc length formula from circular geometry:
$$s = r\theta$$
Where:
Degree to radian conversion:
$$\theta = \frac{\alpha \times \pi}{180}$$
Where $$\alpha$$ is the central angle in degrees.
Combined formula:
$$s = r \cdot \frac{\alpha \pi}{180}$$
Step-by-step process:
The formula follows directly from the definition of radians: one radian is the angle subtended by an arc whose length equals the radius. Therefore, for any angle $$\theta$$ radians, the arc length is simply $$\theta$$ times the radius.
The Arc Length is the curved distance along the circle between the two endpoints of the arc. It is always a positive value, measured in the same linear units as the radius. An arc length of 15.708 with radius 10 and angle 90° means the curved path is about 15.71 units long — exactly one-quarter of the full circumference $$2\pi(10) \approx 62.83$$.
The Angle in Radians is provided as a convenience for further calculations. Many physics and engineering formulas require radians rather than degrees.
Inputs
Results
θ = 90 × π/180 = π/2 ≈ 1.5708 rad. Arc length = 10 × 1.5708 = 15.7080 units. This is exactly one-quarter of the full circumference (2π × 10 ≈ 62.83).
Inputs
Results
θ = 30 × π/180 = π/6 ≈ 0.5236 rad. Arc length = 500 × 0.5236 = 261.80 m. A 30° curve at 500 m radius spans about 262 meters of road.
Arc length is the curved distance along the circle between two points, while chord length is the straight-line distance between those same two points. Arc length is always greater than or equal to chord length; they are equal only for a degenerate arc of zero angle.
The formula $$s = r\theta$$ only works when $$\theta$$ is in radians. This is because one radian is defined as the angle where the arc length equals the radius. Using degrees directly in $$s = r\theta$$ would give incorrect results.
Mathematically, yes — an angle greater than 360° represents more than one full revolution. However, for a physical arc on a single circle, angles are typically between 0° and 360°. This calculator limits the angle to 360° for practical purposes.
Simply multiply: $$s = r\theta$$. No conversion needed. For example, an arc with radius 6 and angle 2 radians has length $$6 \times 2 = 12$$ units.
The fraction is $$\frac{\alpha}{360}$$ when the angle $$\alpha$$ is in degrees, or $$\frac{\theta}{2\pi}$$ when in radians. A 45° arc is $$45/360 = 1/8$$ of the circumference.
Arc length is used in road design (curve lengths), railway engineering (track on curves), CNC machining (tool path lengths), gear design (tooth profiles), and structural engineering (arch dimensions). It is also fundamental to navigation on Earth's spherical surface.
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