# What is this angle in other units?

Converts an angle between degrees, radians, multiples of π, gradians, turns and degrees-minutes-seconds, and finds its coterminal and reference angles, quadrant, type, complement, supplement, and sine, cosine and tangent.

- Page: https://www.acalculator.org/math/angle-calculator
- JSON spec: https://www.acalculator.org/math/angle-calculator.json
- Version: f7b931a0f9f3

## Default answer

Example with the default inputs (Angle 150, Unit Degrees (°)): 150° is 2.617993878 radians.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Angle | The angle, in the unit picked below; negative angles turn clockwise. |
| unit | Unit | The unit of the angle: degrees, radians, multiples of π radians, gradians, or turns. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| degrees | Degrees | The angle in degrees. |
| radians | Radians | The angle in radians: degrees × π ÷ 180. |
| piRadians | Radians as a multiple of π | The angle in radians divided by π: degrees ÷ 180. |
| gradians | Gradians | The angle in gradians: degrees × 10 ÷ 9. |
| turns | Turns | The angle in full turns: degrees ÷ 360. |
| dms | Degrees, minutes, seconds | The angle as whole degrees, minutes and seconds, the seconds rounded half up to 2 decimals. |
| coterminal | Coterminal angle in [0°, 360°) | The angle plus or minus whole turns, between 0° and 360°. |
| reference | Reference angle | The acute angle between the terminal side and the x-axis, from 0° to 90°. |
| quadrant | Quadrant | The quadrant or axis of the terminal side. |
| kind | Type | Zero, acute, right, obtuse, straight, reflex, or full turn, for angles from 0° to 360°. |
| complement | Complement | 90° minus the angle, for angles from 0° to 90°. |
| supplement | Supplement | 180° minus the angle, for angles from 0° to 180°. |
| sin | Sine | The sine of the angle. |
| cos | Cosine | The cosine of the angle. |
| tan | Tangent | The tangent of the angle: sine ÷ cosine; none where the cosine is 0 (90°, 270°). |

## Method

radians = degrees × π ÷ 180; gradians = degrees × 10 ÷ 9; turns = degrees ÷ 360; coterminal = degrees mod 360; reference angle from the quadrant; complement = 90° − angle; supplement = 180° − angle.

## Assumptions

- An angle typed in degrees, multiples of π, gradians or turns is converted exactly; one typed in radians uses π to double precision.
- Negative angles turn clockwise; coterminal angles differ by whole turns of 360°.
- The complement is shown for angles from 0° to 90°, the supplement for angles from 0° to 180°, and the type for angles from 0° to 360°.
- Seconds are rounded half up to 2 decimal places, carrying into minutes and degrees.

## Worked examples

1. x = 150, unit = deg gives radians = 2.617994, piRadians = 0.833333, gradians = 166.666667, dms = 150° 0′ 0″, coterminal = 150, reference = 30, quadrant = Quadrant II, kind = Obtuse, supplement = 30, sin = 0.5, cos = -0.866025, tan = -0.57735. Source: OpenStax Algebra and Trigonometry 2e, section 7.1, Angles (radians = degrees × π ÷ 180; coterminal and reference angles), CC BY 4.0 (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles, retrieved 2026-10-01).
2. x = -405, unit = deg gives coterminal = 315, reference = 45, quadrant = Quadrant IV, turns = -1.125, dms = −405° 0′ 0″. Source: OpenStax Algebra and Trigonometry 2e, section 7.1, Angles: coterminal angles differ by multiples of 360° (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles).
3. x = 12.5125, unit = deg gives dms = 12° 30′ 45″, complement = 77.4875, kind = Acute. Source: NIST Special Publication 811, Appendix B: 1° = 60′, 1′ = 60″ (https://www.nist.gov/pml/special-publication-811).
4. x = 0.5, unit = pi gives degrees = 90, quadrant = On the positive y-axis, kind = Right, sin = 1, cos = 0, complement = 0. Source: OpenStax Algebra and Trigonometry 2e, section 7.1, Angles: π ÷ 2 radians = 90° (https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles).
5. x = 1, unit = rad gives degrees = 57.29578, radians = 1, dms = 57° 17′ 44.81″. Source: NIST Special Publication 811, Appendix B: 1 rad = 57.295 78° (https://www.nist.gov/pml/special-publication-811).

## FAQ

### How do I convert degrees to radians?

Multiply by π and divide by 180. 150° is 150 × π ÷ 180 = 5π ÷ 6 ≈ 2.618 radians. To go back, multiply radians by 180 and divide by π: 1 radian ≈ 57.2958°.

### What is a coterminal angle?

Two angles are coterminal when they end on the same side after turning from the positive x-axis. They differ by whole turns of 360°. The calculator gives the coterminal angle from 0° up to (not including) 360°: −405° is coterminal with −405 + 2 × 360 = 315°.

### What is a reference angle?

It is the acute angle between the angle’s terminal side and the x-axis, from 0° to 90°. In quadrant II it is 180° minus the angle, in quadrant III the angle minus 180°, and in quadrant IV 360° minus the angle. The reference angle of 150° is 30°.

### What are complementary and supplementary angles?

Two angles are complementary when they add up to 90° and supplementary when they add up to 180°. The complement of 12.5125° is 77.4875°; the supplement of 150° is 30°.

### How do I write an angle in degrees, minutes and seconds?

Keep the whole degrees, multiply the decimal part by 60 for minutes, then multiply the decimal part of the minutes by 60 for seconds. 12.5125° is 12° and 0.5125 × 60 = 30.75′, so 12° 30′ 45″.

### What is a gradian?

A unit that divides a right angle into 100 parts, so a full turn is 400 gradians (also called gons). Degrees × 10 ÷ 9 gives gradians.

### What kinds of angles are there?

A zero angle is 0°, an acute angle is between 0° and 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, a straight angle is 180°, a reflex angle is between 180° and 360°, and a full turn is 360°.

## Sources

- OpenStax, Algebra and Trigonometry 2e, section 7.1, Angles (degrees and radians, coterminal angles, reference angles), CC BY 4.0, retrieved 2026-10-01. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-1-angles
- NIST Special Publication 811, Guide for the Use of the International System of Units, Appendix B (degree, minute, second, gon, revolution). https://www.nist.gov/pml/special-publication-811