Calculate arc length for circles, Cartesian curves, parametric and polar equations, 3D space curves, coordinate points, and numerical integration, with step-by-step solutions and a live diagram.
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Gold dots mark the midpoint of each subinterval.
Hover a rectangle for its subinterval, midpoint, integrand value and contribution.
Differences between successive refinements are a practical error indicator, not a guaranteed error bound.
Calculation Error
Radius and angle, with sector area and chord.
x(t) and y(t) curves over any interval.
Curves given as r(θ) in polar form.
Cartesian arc length with every step shown.
Space curves x(t), y(t), and z(t).
Numerical integration with Simpson's rule.
Numerical integration, trapezoidal rule.
Path length through coordinate points.
Ellipse ArcArc length of any ellipse from its axes and angles.
Parabola ArcArc length of any parabola from its equation.
Semicircle ArcArc length of any semicircle from radius or diameter.
Sector ArcArc length of a circular sector from radius and angle.
Midpoint RuleArc length by the midpoint rule, with rectangles and error checks.
The calculator above includes seven modes. Each one matches a standard input format from calculus. Choose the mode whose inputs look like your problem.
| Mode | Inputs needed | When to use |
|---|---|---|
| Circle | Radius r, central angle θ | Circles, sectors, wheels, arcs |
| Cartesian | Function y = f(x), interval [a, b] | Function graphs, y as function of x |
| Parametric | x(t), y(t), interval [a, b] | Motion paths, loops, vertical curves |
| Polar | r(θ), angle interval [α, β] | Spirals, cardioids, polar graphs |
| Numerical | f(x), interval, subdivisions n | Integrals with no closed form |
| Points | List of (x, y) coordinates | GPS tracks, sampled data |
| 3D | x(t), y(t), z(t), interval [a, b] | Space curves, helices, trajectories |
Enter the radius r and the central angle θ. The tool computes arc length with L = rθ, and also reports the sector area and chord length from the same two inputs. Switch between degrees and radians with the DEG/RAD toggle.
Enter a function y = f(x) plus the start and end x-values a and b. The calculator differentiates the function, builds the integrand, and evaluates the definite integral numerically with full steps shown. For the complete derivation, try the arc length calculator with steps.
Enter x(t) and y(t) with the parameter interval. This mode suits paths described by motion over time, where neither coordinate is a direct function of the other. For a dedicated tool, see the parametric arc length calculator.
Enter r as a function of θ with the angle interval. Ideal for spirals, cardioids, and other curves most naturally written in polar form. A standalone polar arc length calculator is also available.
Enter f(x), the interval, and the number of subdivisions n, then pick Simpson's rule or the trapezoidal rule. This mode is useful when the arc length integral has no elementary antiderivative or when you want to compare numerical methods. Dedicated Simpson's rule and trapezoidal rule calculators show the full working.
Paste or type a list of (x, y) points. The calculator sums the straight-line distances between consecutive points, which approximates the path length through sampled data such as GPS tracks. See also the dedicated arc length from points tool.
Enter x(t), y(t), and z(t) with the parameter interval to measure the length of a space curve, for example a helix or a 3D trajectory. The dedicated 3D arc length calculator offers the same computation on its own page.
Every arc length formula below measures the same idea: add up tiny straight segments along the curve. The right formula depends on how the curve is described.
Here r is the radius and θ is the central angle in radians. The arc length carries the same unit as r: a radius in meters gives an arc length in meters. This is the simplest case because a circle has constant curvature.
Use when the curve is y = f(x) between x = a and x = b. The term dy/dx is the slope; where the curve is steep, each dx contributes more length. The limits a and b are x-values, not arc positions.
Use when x and y are both functions of a parameter t, with t running from a to b. This handles curves that loop or go vertical, where y cannot be written as a function of x.
Use when the curve is r = r(θ) with θ from α to β. The r² term accounts for circular sweep while (dr/dθ)² accounts for radial change.
The natural extension of the parametric formula into three dimensions: add the squared z-velocity under the root.
Arc length L follows the curve; chord length c cuts straight across. The arc is always longer, except in the limiting case of a zero angle.
Each example below is worked in full so you can verify every step.
Given: radius r = 5 cm, central angle θ = 60°.
Formula: L = rθ, with θ in radians.
Substitution: L = 5 × 1.0472.
Answer: L ≈ 5.236 cm.
Given: radius r = 10 m, central angle θ = 1.5 rad.
Formula: L = rθ.
Substitution: L = 10 × 1.5 = 15.
Answer: L = 15 m. No conversion is needed because the angle is already in radians.
Given: y = x², interval [0, 1].
Formula: L = ∫₀¹ √(1 + (dy/dx)²) dx.
Calculation: at x = 1 the bracket equals (1/2)√5 + (1/4)ln(2 + √5) ≈ 1.1180 + 0.3609 = 1.4789; at x = 0 it equals 0.
Answer: L ≈ 1.4789 units.
Given: x = 3cos t, y = 3sin t, t from 0 to π/2.
Formula: L = ∫ √((dx/dt)² + (dy/dt)²) dt.
Answer: L = 3π/2 ≈ 4.7124 units, which matches the known quarter-circumference of a radius-3 circle.
Degrees divide a full turn into 360 parts; radians measure angle by arc length on a unit circle, with a full turn equal to 2π radians (about 6.2832). The two units describe the same angles: 180° equals π radians, 90° equals π/2 radians, and 1 radian is about 57.2958°.
The unit matters because the circle formula L = rθ is only correct when θ is in radians. Entering 90 degrees as 90 radians would overstate the arc by a factor of about 57.3. The calculator's DEG/RAD toggle converts for you, but always double-check which unit your problem uses before trusting the output.
Most wrong answers come from setup errors, not from the formula itself. Check this list before trusting a result.
The circle formula needs radians. Entering 90 instead of π/2 inflates the answer about 57 times. Use the DEG/RAD toggle or convert first.
L = rθ uses the radius. If your problem gives the diameter, halve it first or your answer will be exactly double the correct value.
For y = f(x) the limits are x-values; for parametric mode they are t-values; for polar they are angles. Mixing these up measures the wrong segment.
A polar curve needs the polar formula with its r² term, and a parametric curve needs both derivatives. Pick the mode that matches how your curve is written.
Simpson's and trapezoidal rules improve as n grows. If two runs with different n disagree noticeably, increase n until the result stabilizes.
Numerical integration and point-to-point summation are approximations. They converge to the true length but are not exact identities like L = rθ for circles.
Find perimeters of circular sectors, lengths of curved boundaries, and distances along arcs in construction layouts.
Estimate cable and belt lengths around pulleys, pipe bend allowances, and material needed for curved parts.
Compute distances traveled along curved trajectories and relate arc position to angular motion.
Measure curved facades, arched openings, and winding walkway distances from plan drawings.
Compute the travel distance of a robot arm tip along a programmed curved path to plan timing and energy use.
Practice setting up and evaluating arc length integrals, the classic application of integration to geometry.
Parameterize curves by arc length for constant-speed animation and evenly spaced sampling along paths.
An arc length calculator measures the distance along a curve between two points. You enter the curve definition (for example radius and angle, or a function with interval limits) and it returns the length, usually with step-by-step working.
For a circle, multiply radius by the central angle in radians (L = rθ). For other curves, use the matching integral formula, or enter the function into the calculator above and pick the right mode.
The circle formula is L = rθ. Cartesian curves use L = ∫√(1 + (dy/dx)²) dx, parametric curves use L = ∫√((dx/dt)² + (dy/dt)²) dt, polar curves use L = ∫√(r² + (dr/dθ)²) dθ, and 3D curves add the z-term under the root.
For circles you need the radius (or diameter, which is twice the radius). For general curves you do not need a radius at all: the Cartesian, parametric, polar, and points modes work from functions or coordinates instead.
Use radians in the formula L = rθ. If your angle is in degrees, convert first (radians = degrees × π/180) or use the calculator's DEG/RAD toggle, which handles the conversion automatically.
Arc length follows the curve; chord length is the straight line between the arc's endpoints. The chord is always shorter (c = 2r·sin(θ/2) for a circle). See the comparison diagram in the formula section above.
Enter x(t) and y(t) with the parameter interval into the Parametric mode. The calculator evaluates ∫√((dx/dt)² + (dy/dt)²) dt over your interval and shows the steps.
Yes. Enter x(t), y(t), and z(t) into the 3D mode. The formula extends the parametric version with the z-derivative: L = ∫√((dx/dt)² + (dy/dt)² + (dz/dt)²) dt.
The result uses the same length unit as your inputs. If the radius is in centimeters, the arc length is in centimeters. Angles themselves are unitless once converted to radians.
Numerical methods (Simpson's, trapezoidal) approximate the integral. Increase the number of subdivisions n for a closer result, and compare against a known case to confirm your setup.