Arc Length of a Space Curve Notes

Published on: September 29, 2026

This Arc Length of a Space Curve Calculator measures how far it is along a curve in space between two parameter values, and shows the working line by line. The derivative of each component gives the speed along that axis, and the length of the velocity vector is the overall speed. Integrating that speed over the interval adds up the distance travelled. Following the same pattern every time keeps speed and distance clearly apart.

Step-by-step method

  1. Set up the problem.
  2. Differentiate each component to get r'(t).
  3. Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).
  4. Integrate the speed over the interval.
  5. Evaluate to get the arc length.

Formula:

\(L = \int_{a}^{b} \sqrt{\left(x'\right)^2 + \left(y'\right)^2 + \left(z'\right)^2}\,dt\)
See Example 1 Hide Example 1

Problem

\(\mathbf{r}(t) = \langle \cos{\left(t \right)},\sin{\left(t \right)},t \rangle,\; t \in [0,2 \pi]\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Differentiate each component to get r'(t).
  3. Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).
  4. Integrate the speed over the interval.
  5. Evaluate to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\mathbf{r}(t) = \langle \cos{\left(t \right)},\sin{\left(t \right)},t \rangle,\; t \in [0,2 \pi]\)

Step 2

Step 2 - Differentiate each component to get r'(t).

In this problem:

\(\mathbf{r}'(t) = \langle - \sin{\left(t \right)},\cos{\left(t \right)},1 \rangle\)

Step 3

Step 3 - Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).

In this problem:

\(\lVert \mathbf{r}'(t) \rVert = \sqrt{2}\)

Step 4

Step 4 - Integrate the speed over the interval.

In this problem:

\(L = \int_{0}^{2 \pi} \sqrt{2}\,dt\)

Step 5

Step 5 - Evaluate to get the arc length.

In this problem:

\(L = 2 \sqrt{2} \pi \approx 8.89\)

Final Answer

\(L = 2 \sqrt{2} \pi \approx 8.89\)
See Example 2 Hide Example 2

Problem

\(\mathbf{r}(t) = \langle t,t,t \rangle,\; t \in [0,1]\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Differentiate each component to get r'(t).
  3. Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).
  4. Integrate the speed over the interval.
  5. Evaluate to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\mathbf{r}(t) = \langle t,t,t \rangle,\; t \in [0,1]\)

Step 2

Step 2 - Differentiate each component to get r'(t).

In this problem:

\(\mathbf{r}'(t) = \langle 1,1,1 \rangle\)

Step 3

Step 3 - Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).

In this problem:

\(\lVert \mathbf{r}'(t) \rVert = \sqrt{3}\)

Step 4

Step 4 - Integrate the speed over the interval.

In this problem:

\(L = \int_{0}^{1} \sqrt{3}\,dt\)

Step 5

Step 5 - Evaluate to get the arc length.

In this problem:

\(L = \sqrt{3} \approx 1.73\)

Final Answer

\(L = \sqrt{3} \approx 1.73\)
See Example 3 Hide Example 3

Problem

\(\mathbf{r}(t) = \langle 2 t,t,2 t \rangle,\; t \in [0,3]\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Differentiate each component to get r'(t).
  3. Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).
  4. Integrate the speed over the interval.
  5. Evaluate to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\mathbf{r}(t) = \langle 2 t,t,2 t \rangle,\; t \in [0,3]\)

Step 2

Step 2 - Differentiate each component to get r'(t).

In this problem:

\(\mathbf{r}'(t) = \langle 2,1,2 \rangle\)

Step 3

Step 3 - Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).

In this problem:

\(\lVert \mathbf{r}'(t) \rVert = 3\)

Step 4

Step 4 - Integrate the speed over the interval.

In this problem:

\(L = \int_{0}^{3} 3\,dt\)

Step 5

Step 5 - Evaluate to get the arc length.

In this problem:

\(L = 9\)

Final Answer

\(L = 9\)
See Example 4 Hide Example 4

Problem

\(\mathbf{r}(t) = \langle 3 \cos{\left(t \right)},3 \sin{\left(t \right)},4 t \rangle,\; t \in [0,\pi]\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Differentiate each component to get r'(t).
  3. Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).
  4. Integrate the speed over the interval.
  5. Evaluate to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem:

\(\mathbf{r}(t) = \langle 3 \cos{\left(t \right)},3 \sin{\left(t \right)},4 t \rangle,\; t \in [0,\pi]\)

Step 2

Step 2 - Differentiate each component to get r'(t).

In this problem:

\(\mathbf{r}'(t) = \langle - 3 \sin{\left(t \right)},3 \cos{\left(t \right)},4 \rangle\)

Step 3

Step 3 - Form the speed |r'(t)| = sqrt(x'^2 + y'^2 + z'^2).

In this problem:

\(\lVert \mathbf{r}'(t) \rVert = 5\)

Step 4

Step 4 - Integrate the speed over the interval.

In this problem:

\(L = \int_{0}^{\pi} 5\,dt\)

Step 5

Step 5 - Evaluate to get the arc length.

In this problem:

\(L = 5 \pi \approx 15.71\)

Final Answer

\(L = 5 \pi \approx 15.71\)