How is the arc length worked out?

For y = f(x), length is the integral of sqrt(1 + f'(x)^2); for a parametric curve it is the integral of sqrt(x'(t)^2 + y'(t)^2). The page forms symbolic derivatives, then numerically integrates the speed with adaptive Simpson panels. It compares a panel with its halves and with a split at one third to reduce sampling aliasing. The displayed error is an estimate rounded up to a power of ten, with a floor of 1e-12; the default parabola on 0 to 1 reports < 1e-9. This is not a rigorous error bound for every possible input.

Why does it show an error estimate instead of an exact answer?

Many curve-length integrals have no simple elementary antiderivative, and even a closed-form value is displayed numerically here. The curve readout reports y = x on 0 to 1 as 1.414214, approximating sqrt(2), and a quarter of the unit circle as 1.570796, approximating pi/2. The error estimate helps assess the integration, but does not certify every displayed digit. The separate circle calculator uses direct formulas and labels its rounded values separately.

Why does 1/x across zero fail instead of giving a number?

The speed of 1/x is not finite at zero and the curve has no finite length across that pole. This numerical integrator refuses nonfinite speeds at sampled points. It also refuses the endpoint derivative of y = sqrt(1 - x^2) at -1 and 1, although that semicircle has finite length: an infinite derivative does not always mean infinite arc length. Use the circle calculator or a regular parametric description for that semicircle; excluding a singular endpoint measures only the shortened interval.

How do I know the number is not quietly wrong?

The readout shows the integration error estimate and a sampled polyline length. A line through 513 points on a continuous curve is a lower bound for its length, so a shorter integral result is refused. A second line through 2,049 points supplies a convergence check; apparent convergence of sampled lines is a heuristic, not proof that a narrow feature was captured. For the astroid x = cos(t)^3, y = sin(t)^3 over 0 to 2pi, the existing checks return 6.000000 rather than the zero that one poorly placed Simpson panel could suggest.

Can it measure a parametric curve?

Yes. Switch the Curve control to x(t), y(t) and you get two boxes, one for each coordinate, with the interval running over t rather than x. That mode is what handles curves a function of x cannot describe, such as a full circle: x = cos t, y = sin t from 0 to pi/2 is a quarter of the unit circle and comes to 1.570796. Bounds are read as expressions, so pi/2 and 2pi can be typed as they are written.

What do the tolerance and the evaluation cap do?

The tolerance is how close the answer has to be before adaptive Simpson stops subdividing, and it runs from 1e-3 down to 1e-8. Tighter is not always better: it costs more evaluations of the speed, and below 1e-8 a sum of double precision panels cannot honestly hold the claim anyway. The evaluation cap is the budget, up to 20,000, and it is what stops a curve that oscillates faster than the method can follow from freezing the page. A curve that spends the whole budget without meeting the tolerance says so and suggests loosening the tolerance, raising the cap or splitting the interval, instead of printing whatever it had reached. y = sin(32x) over 0 to 2pi is one of those: at the tightest tolerance it spends the whole budget and refuses, and at 1e-3 it comes back as 128.334460 with a bound of < 1e-4.

Does it agree with the textbook examples?

The standard examples agree to the displayed precision. A flat line y = 0 from 0 to 3 returns 3; the parabola y = x^2 from 0 to 1 returns 1.478943, approximating sqrt(5)/2 + asinh(2)/4. The catenary y = cosh(x) from 0 to 1 returns 1.175201, approximating sinh(1). Ellipses also fit parametric mode: for semi-axes 2 and 1, use x(t) = 2*cos(t), y(t) = sin(t), and bounds 0 to 2pi for the full perimeter.

Does my curve leave the browser?

No. There is no account, no file to send and no server call: the reading, the derivative and the integration all happen in this tab, and nothing you type is sent anywhere or recorded in analytics. The curve, the interval and the settings are kept in your browser local storage so the page opens where you left it, and the Start over button at the top of the page forgets them.

How do I find a circle arc from radius and angle?

In Circle arc, chord and sector, choose Radius and angle. Enter the radius and the measured angle, choose degrees or radians, and state whether it is a central or inscribed angle. A central angle theta in radians gives arc length s = r*theta. For degrees, s = r*angle*pi/180; an inscribed angle is doubled to get the corresponding central angle. Radius 3 and central angle 180 degrees give arc length 3pi, chord 6 and sector area 9pi/2. The pi expression is the formula with your inputs, not a simplified exact answer.

Can I find the radius or angle from an arc length?

Yes. Choose Arc length and angle to solve r = s/theta, or Radius and arc length to solve theta = s/r. The readout includes the radius, diameter, circumference and central angle in both radians and degrees. A positive arc alone cannot determine a radius or angle; the second known measurement is required. This calculator accepts one circular arc, up to a full turn.

How do chord length, major arcs and sector area relate to arc length?

For a circle, chord length is 2r*sin(theta/2) and sector area is r*s/2. Choose Radius and chord length to recover the central angle, then select Minor arc or Major arc because the same chord usually bounds two arcs. A chord cannot exceed the diameter. For central angles greater than 180 degrees, the forward radius-and-angle calculation directly measures the major arc. Chord length is the straight distance between arc endpoints, while arc length follows the circle.

Curve Arc Length Calculator

Q: Why does it show an error estimate instead of an exact answer?

Many curve-length integrals have no simple elementary antiderivative, and even a closed-form value is displayed numerically here. The curve readout reports y = x on 0 to 1 as 1.414214, approximating sqrt(2), and a quarter of the unit circle as 1.570796, approximating pi/2. The error estimate helps assess the integration, but does not certify every displayed digit. The separate circle calculator uses direct formulas and labels its rounded values separately.

Q: Why does 1/x across zero fail instead of giving a number?

The speed of 1/x is not finite at zero and the curve has no finite length across that pole. This numerical integrator refuses nonfinite speeds at sampled points. It also refuses the endpoint derivative of y = sqrt(1 - x^2) at -1 and 1, although that semicircle has finite length: an infinite derivative does not always mean infinite arc length. Use the circle calculator or a regular parametric description for that semicircle; excluding a singular endpoint measures only the shortened interval.

Q: How do I know the number is not quietly wrong?

The readout shows the integration error estimate and a sampled polyline length. A line through 513 points on a continuous curve is a lower bound for its length, so a shorter integral result is refused. A second line through 2,049 points supplies a convergence check; apparent convergence of sampled lines is a heuristic, not proof that a narrow feature was captured. For the astroid x = cos(t)^3, y = sin(t)^3 over 0 to 2pi, the existing checks return 6.000000 rather than the zero that one poorly placed Simpson panel could suggest.

Q: Can it measure a parametric curve?

Yes. Switch the Curve control to x(t), y(t) and you get two boxes, one for each coordinate, with the interval running over t rather than x. That mode is what handles curves a function of x cannot describe, such as a full circle: x = cos t, y = sin t from 0 to pi/2 is a quarter of the unit circle and comes to 1.570796. Bounds are read as expressions, so pi/2 and 2pi can be typed as they are written.

Q: What do the tolerance and the evaluation cap do?

The tolerance is how close the answer has to be before adaptive Simpson stops subdividing, and it runs from 1e-3 down to 1e-8. Tighter is not always better: it costs more evaluations of the speed, and below 1e-8 a sum of double precision panels cannot honestly hold the claim anyway. The evaluation cap is the budget, up to 20,000, and it is what stops a curve that oscillates faster than the method can follow from freezing the page. A curve that spends the whole budget without meeting the tolerance says so and suggests loosening the tolerance, raising the cap or splitting the interval, instead of printing whatever it had reached. y = sin(32x) over 0 to 2pi is one of those: at the tightest tolerance it spends the whole budget and refuses, and at 1e-3 it comes back as 128.334460 with a bound of < 1e-4.

Q: Does it agree with the textbook examples?

The standard examples agree to the displayed precision. A flat line y = 0 from 0 to 3 returns 3; the parabola y = x^2 from 0 to 1 returns 1.478943, approximating sqrt(5)/2 + asinh(2)/4. The catenary y = cosh(x) from 0 to 1 returns 1.175201, approximating sinh(1). Ellipses also fit parametric mode: for semi-axes 2 and 1, use x(t) = 2*cos(t), y(t) = sin(t), and bounds 0 to 2pi for the full perimeter.

Q: Does my curve leave the browser?

No. There is no account, no file to send and no server call: the reading, the derivative and the integration all happen in this tab, and nothing you type is sent anywhere or recorded in analytics. The curve, the interval and the settings are kept in your browser local storage so the page opens where you left it, and the Start over button at the top of the page forgets them.

Q: How do I find a circle arc from radius and angle?

In Circle arc, chord and sector, choose Radius and angle. Enter the radius and the measured angle, choose degrees or radians, and state whether it is a central or inscribed angle. A central angle theta in radians gives arc length s = r*theta. For degrees, s = r*angle*pi/180; an inscribed angle is doubled to get the corresponding central angle. Radius 3 and central angle 180 degrees give arc length 3pi, chord 6 and sector area 9pi/2. The pi expression is the formula with your inputs, not a simplified exact answer.

Q: Can I find the radius or angle from an arc length?

Yes. Choose Arc length and angle to solve r = s/theta, or Radius and arc length to solve theta = s/r. The readout includes the radius, diameter, circumference and central angle in both radians and degrees. A positive arc alone cannot determine a radius or angle; the second known measurement is required. This calculator accepts one circular arc, up to a full turn.

Type a curve and an interval and the arc length appears with its own error estimate underneath. Measure y = f(x) or x(t), y(t) between two bounds, with symbolic derivatives and adaptive Simpson integration. The readout shows an estimated numerical error and compares the result with lines through 513 and 2,049 sampled curve points. For a circular arc, use the separate two-measurement calculator below for radius, angle, chord and sector area. Numerical estimates and sampling checks are useful checks, not a proof of exactness.

Arc length1.478943y = x^2 on 0 to 1 · Adaptive Simpson · estimated error < 1e-9

Curve

y =

x from

Tolerance

Evaluation cap

Length: 1.478943
Interval: 0 to 1
Est. error: < 1e-9
Method: Adaptive Simpson

Adaptive Simpson on a speed of sqrt(1+(2*x)^2), accepted in 32 segments after 240 evaluations, under a tolerance of 1e-8.

dy/dx = 2*x

The line through 513 points on the curve is 1.478943 long, and an arc is never shorter than that.

y = x on 0 to 1a straight line
1.414214, which is √2
y = x² on 0 to 1curved
1.478943
x = cos t, y = sin t on 0 to π/2parametric
1.570796, which is π/2

Circle arc, chord and sector

Use two known measurements for one circular arc. Length inputs use the same unit; sector area uses that unit squared. These fields are separate from the curve above and reset when you reload.

Known measurementsRadiusAngleAngle unitAngle type

Circle calculated. Displayed numbers are rounded; the pi expression is a formula, not a simplified exact result.

Circular arc length: 9.424777961
Radius: 3
Diameter: 6
Circumference: 18.84955592
Central angle (radians): 3.141592654
Central angle (degrees): 180
Inscribed angle (degrees): 90
Chord length: 6
Sector area: 14.13716694

s = (3) * (180) * pi / 180
chord = 2r * sin(theta/2); sector area = r * s / 2

Accuracy. The length is a numerical approximation and never an exact one, so the page prints its own error estimate beside it and you can see how many digits to trust. Integration is adaptive Simpson on the speed of the curve, under the tolerance and evaluation cap you set. An interval containing a singularity, such as 1/x across zero, fails by name instead of returning a number that looks like an answer.

What you can type

Numbers, the variable of the mode, the constants pi and e, the signs + - * / ^ and brackets, and these functions: sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, sqrt, abs, log. ln is read as log, which is the natural logarithm here, and arcsin, arccos and arctan are read as asin, acos and atan. Multiplication can be left out where it is obvious, so 2x, 3 cos(t) and 2(x + 1) all read the way you would write them on paper, and the stage prints back what that came to. The bounds take the same expressions, so pi/2 and 2pi are intervals. An expression runs to 500 characters, the tolerance from 1e-3 to 1e-8, and the evaluation cap from 300 to 20000.

Common questions

How is the arc length worked out?: For y = f(x), length is the integral of sqrt(1 + f'(x)^2); for a parametric curve it is the integral of sqrt(x'(t)^2 + y'(t)^2). The page forms symbolic derivatives, then numerically integrates the speed with adaptive Simpson panels. It compares a panel with its halves and with a split at one third to reduce sampling aliasing. The displayed error is an estimate rounded up to a power of ten, with a floor of 1e-12; the default parabola on 0 to 1 reports < 1e-9. This is not a rigorous error bound for every possible input.
Why does it show an error estimate instead of an exact answer?: Many curve-length integrals have no simple elementary antiderivative, and even a closed-form value is displayed numerically here. The curve readout reports y = x on 0 to 1 as 1.414214, approximating sqrt(2), and a quarter of the unit circle as 1.570796, approximating pi/2. The error estimate helps assess the integration, but does not certify every displayed digit. The separate circle calculator uses direct formulas and labels its rounded values separately.
Why does 1/x across zero fail instead of giving a number?: The speed of 1/x is not finite at zero and the curve has no finite length across that pole. This numerical integrator refuses nonfinite speeds at sampled points. It also refuses the endpoint derivative of y = sqrt(1 - x^2) at -1 and 1, although that semicircle has finite length: an infinite derivative does not always mean infinite arc length. Use the circle calculator or a regular parametric description for that semicircle; excluding a singular endpoint measures only the shortened interval.
How do I know the number is not quietly wrong?: The readout shows the integration error estimate and a sampled polyline length. A line through 513 points on a continuous curve is a lower bound for its length, so a shorter integral result is refused. A second line through 2,049 points supplies a convergence check; apparent convergence of sampled lines is a heuristic, not proof that a narrow feature was captured. For the astroid x = cos(t)^3, y = sin(t)^3 over 0 to 2pi, the existing checks return 6.000000 rather than the zero that one poorly placed Simpson panel could suggest.
Can it measure a parametric curve?: Yes. Switch the Curve control to x(t), y(t) and you get two boxes, one for each coordinate, with the interval running over t rather than x. That mode is what handles curves a function of x cannot describe, such as a full circle: x = cos t, y = sin t from 0 to pi/2 is a quarter of the unit circle and comes to 1.570796. Bounds are read as expressions, so pi/2 and 2pi can be typed as they are written.
What do the tolerance and the evaluation cap do?: The tolerance is how close the answer has to be before adaptive Simpson stops subdividing, and it runs from 1e-3 down to 1e-8. Tighter is not always better: it costs more evaluations of the speed, and below 1e-8 a sum of double precision panels cannot honestly hold the claim anyway. The evaluation cap is the budget, up to 20,000, and it is what stops a curve that oscillates faster than the method can follow from freezing the page. A curve that spends the whole budget without meeting the tolerance says so and suggests loosening the tolerance, raising the cap or splitting the interval, instead of printing whatever it had reached. y = sin(32x) over 0 to 2pi is one of those: at the tightest tolerance it spends the whole budget and refuses, and at 1e-3 it comes back as 128.334460 with a bound of < 1e-4.
Does it agree with the textbook examples?: The standard examples agree to the displayed precision. A flat line y = 0 from 0 to 3 returns 3; the parabola y = x^2 from 0 to 1 returns 1.478943, approximating sqrt(5)/2 + asinh(2)/4. The catenary y = cosh(x) from 0 to 1 returns 1.175201, approximating sinh(1). Ellipses also fit parametric mode: for semi-axes 2 and 1, use x(t) = 2*cos(t), y(t) = sin(t), and bounds 0 to 2pi for the full perimeter.
Does my curve leave the browser?: No. There is no account, no file to send and no server call: the reading, the derivative and the integration all happen in this tab, and nothing you type is sent anywhere or recorded in analytics. The curve, the interval and the settings are kept in your browser local storage so the page opens where you left it, and the Start over button at the top of the page forgets them.
How do I find a circle arc from radius and angle?: In Circle arc, chord and sector, choose Radius and angle. Enter the radius and the measured angle, choose degrees or radians, and state whether it is a central or inscribed angle. A central angle theta in radians gives arc length s = r*theta. For degrees, s = r*angle*pi/180; an inscribed angle is doubled to get the corresponding central angle. Radius 3 and central angle 180 degrees give arc length 3pi, chord 6 and sector area 9pi/2. The pi expression is the formula with your inputs, not a simplified exact answer.
Can I find the radius or angle from an arc length?: Yes. Choose Arc length and angle to solve r = s/theta, or Radius and arc length to solve theta = s/r. The readout includes the radius, diameter, circumference and central angle in both radians and degrees. A positive arc alone cannot determine a radius or angle; the second known measurement is required. This calculator accepts one circular arc, up to a full turn.
How do chord length, major arcs and sector area relate to arc length?: For a circle, chord length is 2r*sin(theta/2) and sector area is r*s/2. Choose Radius and chord length to recover the central angle, then select Minor arc or Major arc because the same chord usually bounds two arcs. A chord cannot exceed the diameter. For central angles greater than 180 degrees, the forward radius-and-angle calculation directly measures the major arc. Chord length is the straight distance between arc endpoints, while arc length follows the circle.

The length is a numerical approximation and never an exact one, so the page prints its own error estimate beside it and you can see how many digits to trust. Integration is adaptive Simpson on the speed of the curve, under the tolerance and evaluation cap you set. An interval containing a singularity, such as 1/x across zero, fails by name instead of returning a number that looks like an answer.