Radians to Degrees Notes

Published on: September 29, 2026

This Radians to Degrees Calculator rewrites an angle measured in radians as the same angle measured in degrees, and shows the working line by line. A half turn is one pi radians and also 180 degrees, so multiplying by 180 over pi undoes the other conversion. The pi on the top cancels the pi on the bottom, which is why so many familiar angles land on whole numbers. 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 radians into the formula.
  4. Cancel and simplify to get the angle in degrees.

Formula:

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

Problem

\(\theta = \frac{\pi}{6}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in radians into the formula.
  4. Cancel and simplify to get the angle in degrees.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(\frac{\pi}{6}\) radians, and it has to end up measured in degrees.

\(\theta = \frac{\pi}{6}\)

Step 2

Step 2 - Write the conversion formula.

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

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

Step 3

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

In this problem: Replacing \(\text{radians}\) with \(\frac{\pi}{6}\) gives \(\text{degrees} = \frac{\pi}{6} \times \frac{180}{\pi}\).

\(\text{degrees} = \frac{\pi}{6} \times \frac{180}{\pi}\)

Step 4

Step 4 - Cancel and simplify to get the angle in degrees.

In this problem: The \(\pi\) on the top cancels the \(\pi\) on the bottom, and what is left works out to \(30^{\circ}\).

\(\text{degrees} = 30^{\circ}\)

Final Answer

\(\text{degrees} = 30^{\circ}\)
See Example 2 Hide Example 2

Problem

\(\theta = \frac{2\pi}{3}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in radians into the formula.
  4. Cancel and simplify to get the angle in degrees.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(\frac{2\pi}{3}\) radians, and it has to end up measured in degrees.

\(\theta = \frac{2\pi}{3}\)

Step 2

Step 2 - Write the conversion formula.

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

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

Step 3

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

In this problem: Replacing \(\text{radians}\) with \(\frac{2\pi}{3}\) gives \(\text{degrees} = \frac{2\pi}{3} \times \frac{180}{\pi}\).

\(\text{degrees} = \frac{2\pi}{3} \times \frac{180}{\pi}\)

Step 4

Step 4 - Cancel and simplify to get the angle in degrees.

In this problem: The \(\pi\) on the top cancels the \(\pi\) on the bottom, and what is left works out to \(120^{\circ}\).

\(\text{degrees} = 120^{\circ}\)

Final Answer

\(\text{degrees} = 120^{\circ}\)
See Example 3 Hide Example 3

Problem

\(\theta = \frac{5\pi}{6}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in radians into the formula.
  4. Cancel and simplify to get the angle in degrees.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(\frac{5\pi}{6}\) radians, and it has to end up measured in degrees.

\(\theta = \frac{5\pi}{6}\)

Step 2

Step 2 - Write the conversion formula.

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

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

Step 3

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

In this problem: Replacing \(\text{radians}\) with \(\frac{5\pi}{6}\) gives \(\text{degrees} = \frac{5\pi}{6} \times \frac{180}{\pi}\).

\(\text{degrees} = \frac{5\pi}{6} \times \frac{180}{\pi}\)

Step 4

Step 4 - Cancel and simplify to get the angle in degrees.

In this problem: The \(\pi\) on the top cancels the \(\pi\) on the bottom, and what is left works out to \(150^{\circ}\).

\(\text{degrees} = 150^{\circ}\)

Final Answer

\(\text{degrees} = 150^{\circ}\)
See Example 4 Hide Example 4

Problem

\(\theta = \frac{3\pi}{2}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the conversion formula.
  3. Substitute the angle in radians into the formula.
  4. Cancel and simplify to get the angle in degrees.

Step 1

Step 1 - Set up the problem.

In this problem: The angle to convert is \(\frac{3\pi}{2}\) radians, and it has to end up measured in degrees.

\(\theta = \frac{3\pi}{2}\)

Step 2

Step 2 - Write the conversion formula.

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

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

Step 3

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

In this problem: Replacing \(\text{radians}\) with \(\frac{3\pi}{2}\) gives \(\text{degrees} = \frac{3\pi}{2} \times \frac{180}{\pi}\).

\(\text{degrees} = \frac{3\pi}{2} \times \frac{180}{\pi}\)

Step 4

Step 4 - Cancel and simplify to get the angle in degrees.

In this problem: The \(\pi\) on the top cancels the \(\pi\) on the bottom, and what is left works out to \(270^{\circ}\).

\(\text{degrees} = 270^{\circ}\)

Final Answer

\(\text{degrees} = 270^{\circ}\)