MathematicsCalculusA-Level
CBSEGCE A-LevelAbiturAPBaccalauréat GénéralCambridgeCAPSCCEA

Arc Length (Parametric) Calculator

Calculate the length of a parametric curve.

Use the free calculatorCheck the variablesOpen the advanced solver
Open Advanced Calculator
This is the free calculator preview. Advanced walkthroughs stay in the app.
Result
Ready
Arc length

Formula first

Overview

The parametric arc length formula calculates the total distance along a path where coordinates are defined as separate functions of a shared parameter, usually time. It sums infinitesimal segments of the curve by integrating the magnitude of the velocity vector over the specified interval.

Symbols

Variables

R = Radius / speed, a = Start parameter a, b = End parameter b, L = Arc length

Radius / speed
Variable
Start parameter a
rad
End parameter b
rad
Arc length
Variable

Apply it well

When To Use

When to use: Apply this formula when a curve is defined by x(t) and y(t) rather than a direct relationship between x and y. It is required that the derivatives of these functions are continuous and that the path is not retraced during the integration interval.

Why it matters: This is a fundamental tool in physics for calculating the total distance traveled by objects in motion, such as satellites or projectiles. In manufacturing, it helps determine the exact length of material needed to form curved components in engineering designs.

Avoid these traps

Common Mistakes

  • Forgetting square root.
  • Integrating x(t) instead of derivatives.

One free problem

Practice Problem

A particle moves along a circular path defined by x = 5 cos(t) and y = 5 sin(t). Calculate the total distance traveled by the particle as the parameter t goes from 0 to 2π.

Start parameter a0 rad
End parameter b6.2831853 rad
Radius / speed5

Solve for:

Hint: The square root of the sum of the squared derivatives simplifies to the radius of the circle.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Stewart, James. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2016.
  2. Thomas, George B. Jr., Maurice D. Weir, and Joel Hass. Thomas' Calculus. 14th ed. Pearson, 2018.
  3. Wikipedia: Arc length
  4. Stewart, James. Calculus: Early Transcendentals. 8th ed. Cengage Learning, 2015.
  5. Halliday, David, Robert Resnick, and Jearl Walker. Fundamentals of Physics. 11th ed. Wiley, 2018.
  6. Stewart, James. Calculus: Early Transcendentals. 8th ed., Cengage Learning, 2016.
  7. Thomas, George B., et al. Thomas' Calculus. 14th ed., Pearson, 2018.
  8. Edexcel Further Mathematics — Core Pure (Calculus)