Arc Length Notes

Published on: September 29, 2026

This Arc Length Calculator measures how far it is along the curved edge of a circle between the two ends of an angle, and shows the working line by line. The formula only behaves when the angle is in radians, because a radian counts how many radius lengths fit along the arc, so the angle is converted first. After that the arc length is a single multiplication. Following the same pattern every time stops the conversion being forgotten.

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the arc length formula.
  4. Substitute the radius and the angle in radians.
  5. Multiply to get the arc length.

Formulas:

\(L = r \times \theta_{rad}\)
\(\theta_{rad} = \theta \times \frac{\pi}{180}\)
See Example 1 Hide Example 1

Problem

\(r = 7,\; \theta = 45^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the arc length formula.
  4. Substitute the radius and the angle in radians.
  5. Multiply to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem: The circle has radius \(r = 7\) and the arc turns through \(\theta = 45^{\circ}\).

\(r = 7,\; \theta = 45^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(45^{\circ}\) by \(\frac{\pi}{180}\) gives \(0.79\) radians.

\(\theta_{rad} = \frac{45\pi}{180} \approx 0.79\)

Step 3

Step 3 - Write the arc length formula.

In this problem: An angle in radians already counts how many radius lengths fit along the arc, so the arc length is \(L = r \times \theta_{rad}\).

\(L = r \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(7\) and \(\theta_{rad}\) with \(0.79\) gives \(L = 7 \times 0.79\).

\(L = 7 \times 0.79\)

Step 5

Step 5 - Multiply to get the arc length.

In this problem: Multiplying \(7\) by \(0.79\) gives \(5.5\).

\(L = 7 \times 0.79 \approx 5.5\)

Final Answer

\(L \approx 5.5\)
See Example 2 Hide Example 2

Problem

\(r = 3,\; \theta = 120^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the arc length formula.
  4. Substitute the radius and the angle in radians.
  5. Multiply to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem: The circle has radius \(r = 3\) and the arc turns through \(\theta = 120^{\circ}\).

\(r = 3,\; \theta = 120^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(120^{\circ}\) by \(\frac{\pi}{180}\) gives \(2.09\) radians.

\(\theta_{rad} = \frac{120\pi}{180} \approx 2.09\)

Step 3

Step 3 - Write the arc length formula.

In this problem: An angle in radians already counts how many radius lengths fit along the arc, so the arc length is \(L = r \times \theta_{rad}\).

\(L = r \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(3\) and \(\theta_{rad}\) with \(2.09\) gives \(L = 3 \times 2.09\).

\(L = 3 \times 2.09\)

Step 5

Step 5 - Multiply to get the arc length.

In this problem: Multiplying \(3\) by \(2.09\) gives \(6.28\).

\(L = 3 \times 2.09 \approx 6.28\)

Final Answer

\(L \approx 6.28\)
See Example 3 Hide Example 3

Problem

\(r = 5,\; \theta = 60^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the arc length formula.
  4. Substitute the radius and the angle in radians.
  5. Multiply to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem: The circle has radius \(r = 5\) and the arc turns through \(\theta = 60^{\circ}\).

\(r = 5,\; \theta = 60^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(60^{\circ}\) by \(\frac{\pi}{180}\) gives \(1.05\) radians.

\(\theta_{rad} = \frac{60\pi}{180} \approx 1.05\)

Step 3

Step 3 - Write the arc length formula.

In this problem: An angle in radians already counts how many radius lengths fit along the arc, so the arc length is \(L = r \times \theta_{rad}\).

\(L = r \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(5\) and \(\theta_{rad}\) with \(1.05\) gives \(L = 5 \times 1.05\).

\(L = 5 \times 1.05\)

Step 5

Step 5 - Multiply to get the arc length.

In this problem: Multiplying \(5\) by \(1.05\) gives \(5.24\).

\(L = 5 \times 1.05 \approx 5.24\)

Final Answer

\(L \approx 5.24\)
See Example 4 Hide Example 4

Problem

\(r = 8,\; \theta = 135^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the arc length formula.
  4. Substitute the radius and the angle in radians.
  5. Multiply to get the arc length.

Step 1

Step 1 - Set up the problem.

In this problem: The circle has radius \(r = 8\) and the arc turns through \(\theta = 135^{\circ}\).

\(r = 8,\; \theta = 135^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(135^{\circ}\) by \(\frac{\pi}{180}\) gives \(2.36\) radians.

\(\theta_{rad} = \frac{135\pi}{180} \approx 2.36\)

Step 3

Step 3 - Write the arc length formula.

In this problem: An angle in radians already counts how many radius lengths fit along the arc, so the arc length is \(L = r \times \theta_{rad}\).

\(L = r \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(8\) and \(\theta_{rad}\) with \(2.36\) gives \(L = 8 \times 2.36\).

\(L = 8 \times 2.36\)

Step 5

Step 5 - Multiply to get the arc length.

In this problem: Multiplying \(8\) by \(2.36\) gives \(18.85\).

\(L = 8 \times 2.36 \approx 18.85\)

Final Answer

\(L \approx 18.85\)