Degrees to Radians Notes

Published on: September 29, 2026

This Degrees to Radians Calculator rewrites an angle measured in degrees as the same angle measured in radians, and shows the working line by line. A half turn is 180 degrees and also one pi radians, which is what turns the conversion into a single multiplication. Whole-number angles usually leave a neat fraction of pi, and awkward ones fall back on a decimal instead. Following the same pattern every time makes the conversion quick to check and easy to repeat.

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in degrees into the formula.
  4. Simplify the result.

Formula:

\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)
See Example 1 Hide Example 1

Problem

\(\theta = 270^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in degrees into the formula.
  4. Simplify the result.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(270^{\circ}\), and it has to end up measured in radians.

\(\theta = 270^{\circ}\)

Step 2

Step 2 - Write the conversion formula.

In this problem: A half turn is \(180^{\circ}\) and also \(\pi\) radians, so every degree is worth \(\frac{\pi}{180}\) radians.

\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

Step 3

Step 3 - Substitute the angle in degrees into the formula.

In this problem: Replacing \(\text{degrees}\) with \(270\) gives \(\text{radians} = \frac{270\pi}{180}\).

\(\text{radians} = \frac{270\pi}{180}\)

Step 4

Step 4 - Simplify the result.

In this problem: Dividing the top and the bottom of \(\frac{270\pi}{180}\) by \(90\) gives \(\frac{3\pi}{2}\).

\(\text{radians} = \frac{3\pi}{2}\)

Final Answer

\(\text{radians} = \frac{3\pi}{2}\)
See Example 2 Hide Example 2

Problem

\(\theta = 135^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in degrees into the formula.
  4. Simplify the result.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(135^{\circ}\), and it has to end up measured in radians.

\(\theta = 135^{\circ}\)

Step 2

Step 2 - Write the conversion formula.

In this problem: A half turn is \(180^{\circ}\) and also \(\pi\) radians, so every degree is worth \(\frac{\pi}{180}\) radians.

\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

Step 3

Step 3 - Substitute the angle in degrees into the formula.

In this problem: Replacing \(\text{degrees}\) with \(135\) gives \(\text{radians} = \frac{135\pi}{180}\).

\(\text{radians} = \frac{135\pi}{180}\)

Step 4

Step 4 - Simplify the result.

In this problem: Dividing the top and the bottom of \(\frac{135\pi}{180}\) by \(45\) gives \(\frac{3\pi}{4}\).

\(\text{radians} = \frac{3\pi}{4}\)

Final Answer

\(\text{radians} = \frac{3\pi}{4}\)
See Example 3 Hide Example 3

Problem

\(\theta = 120^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in degrees into the formula.
  4. Simplify the result.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(120^{\circ}\), and it has to end up measured in radians.

\(\theta = 120^{\circ}\)

Step 2

Step 2 - Write the conversion formula.

In this problem: A half turn is \(180^{\circ}\) and also \(\pi\) radians, so every degree is worth \(\frac{\pi}{180}\) radians.

\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

Step 3

Step 3 - Substitute the angle in degrees into the formula.

In this problem: Replacing \(\text{degrees}\) with \(120\) gives \(\text{radians} = \frac{120\pi}{180}\).

\(\text{radians} = \frac{120\pi}{180}\)

Step 4

Step 4 - Simplify the result.

In this problem: Dividing the top and the bottom of \(\frac{120\pi}{180}\) by \(60\) gives \(\frac{2\pi}{3}\).

\(\text{radians} = \frac{2\pi}{3}\)

Final Answer

\(\text{radians} = \frac{2\pi}{3}\)
See Example 4 Hide Example 4

Problem

\(\theta = 225^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in degrees into the formula.
  4. Simplify the result.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(225^{\circ}\), and it has to end up measured in radians.

\(\theta = 225^{\circ}\)

Step 2

Step 2 - Write the conversion formula.

In this problem: A half turn is \(180^{\circ}\) and also \(\pi\) radians, so every degree is worth \(\frac{\pi}{180}\) radians.

\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)

Step 3

Step 3 - Substitute the angle in degrees into the formula.

In this problem: Replacing \(\text{degrees}\) with \(225\) gives \(\text{radians} = \frac{225\pi}{180}\).

\(\text{radians} = \frac{225\pi}{180}\)

Step 4

Step 4 - Simplify the result.

In this problem: Dividing the top and the bottom of \(\frac{225\pi}{180}\) by \(45\) gives \(\frac{5\pi}{4}\).

\(\text{radians} = \frac{5\pi}{4}\)

Final Answer

\(\text{radians} = \frac{5\pi}{4}\)