Angle Converter

About Angle Converter

Angle conversion: 360° = 2π radians = 400 gradians = 1 turn.

How to use this calculator

1. Enter the angle value you already know.
2. Choose the unit you are converting from, such as degrees, radians, gradians, or turns.
3. Select the unit you want to convert to.
4. Read the converted result and, if needed, use the step-by-step breakdown to check the calculation.

The formula explained

The key relationship is \(360^\circ = 2\pi\text{ rad} = 400\text{ grad} = 1\text{ turn}\). This means every angle unit can be converted by comparing it to the same full rotation.

• \(\theta\) = the angle value being converted
• \(\circ\) = degrees
• \(\text{rad}\) = radians
• \(\text{grad}\) = gradians
• \(\text{turn}\) = turns, where \(1\text{ turn}\) is one full rotation

Step by step method

1. Write the angle in a known unit, for example \(45^\circ\).
2. Use the full-rotation equivalence \(360^\circ = 2\pi\text{ rad} = 400\text{ grad} = 1\text{ turn}\) to build a conversion factor.
3. Multiply by the fraction that cancels the original unit and leaves the target unit.
4. Simplify the result, and if needed round to the requested decimal places.

Worked example

Problem. Convert \(135^\circ\) to radians, gradians, and turns.

1. For radians, use \(135^\circ \times \frac{2\pi\text{ rad}}{360^\circ} = \frac{135\pi}{180}\text{ rad} = \frac{3\pi}{4}\text{ rad}\).
2. For gradians, use \(135^\circ \times \frac{400\text{ grad}}{360^\circ} = 150\text{ grad}\).
3. For turns, use \(135^\circ \times \frac{1\text{ turn}}{360^\circ} = \frac{3}{8}\text{ turn}\).

Tips and common mistakes

• Make sure the unit you start with appears in the denominator of your conversion factor, so it cancels correctly.
• If you convert to radians, leave answers in terms of \(\pi\) when possible, because that is often the exact form used in math.

Frequently asked questions