# Angle Converter - Degrees, Radians, DMS & Unit Circle

> Convert degrees ↔ radians (exact, as π/6 or 3π/4) ↔ DMS ↔ gradians ↔ turns, find coterminal and reference angles, and drag an interactive unit circle.

- URL: https://qilingtools.com/en/math/angle
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Converters](https://qilingtools.com/en/tags/convert)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter an angle, choose its unit — degrees, radians, degrees-minutes-seconds, gradians (gon) or turns — and see it in every other unit at once. Any angle that is a rational number of degrees is converted exactly with fractions: 30° becomes π/6, 135° becomes 3π/4, 12.5° becomes 5π/72 and DMS reads 30° 15′ 30″, instead of a bare 0.5235987756. Radians can be typed as pi/6, 3π/4 or 2pi. You also get the coterminal angle in [0°, 360°) and in (−180°, 180°], the quadrant (or the axis the angle lies on), the reference angle, a list of coterminal angles and the general form θ + k·360°. The interactive unit circle lets you drag the point around (with optional snapping to multiples of 15°), draws cos, sin and tan as segments, and gives exact values for multiples of 15° and 18°, such as sin 15° = (√6 − √2)/4.

## How to use

1. Pick the input unit and type an angle, e.g. 30, -210, pi/6, 30°15′30″ or 1/8 (turn)
2. Read the conversions, coterminal angles, quadrant and reference angle, and copy any value
3. Drag the point on the unit circle to change the angle and see exact sin, cos and tan

## Features

- Converts between degrees, radians, DMS, gradians and turns
- Exact fraction arithmetic for rational angles; radians shown as fractions of π
- Radian input accepts expressions like pi/6, 3π/4 or 2pi
- Normalisation to [0°, 360°) and (−180°, 180°], coterminal angles and general form
- Quadrant or quadrantal-axis detection and the reference angle
- Draggable unit circle with special-angle snapping and sin, cos, tan segments
- Exact surd values of sin, cos and tan for multiples of 15° and 18°
- One-click copy and a link to the trig calculator

## FAQ

### How do I convert degrees to radians?

Since 180° = π radians, multiply degrees by π/180 to get radians and multiply radians by 180/π to get degrees. So 30° = π/6, 135° = 3π/4 and 1 rad ≈ 57.2958°. The converter reduces the fraction exactly for whole, decimal and fractional degrees, and only falls back to decimals for angles like 1.2 rad that are not a rational multiple of π.

### How do degrees-minutes-seconds convert to decimal degrees?

One degree is 60 minutes and one minute is 60 seconds, so decimal degrees = D + M/60 + S/3600. For example 30°15′30″ = 30 + 15/60 + 30/3600 = 30.258333…°. Going the other way, multiply the decimal part by 60 for minutes and the remaining fraction by 60 again for seconds. You can type 30°15′30″, 30 15 30 or 30d15m30s.

### What are coterminal and reference angles?

Coterminal angles share the same terminal side and differ by whole turns: θ + k·360° (or θ + 2kπ), so −210°, 150° and 510° are coterminal. The reference angle is the acute angle between the terminal side and the x-axis: θ in quadrant I, 180° − θ in II, θ − 180° in III and 360° − θ in IV. A trig function of any angle equals the same function of its reference angle with the sign set by the quadrant.

### What are gradians and turns?

Gradians (gon) split a full circle into 400 parts, so a right angle is 100 gon and 1 gon = 0.9°; they are used mainly in surveying. A turn is one full revolution, 360° or 2π, handy for rotation counts. Both convert exactly, e.g. 50 gon = 45° = π/4 and 1/8 turn = 45°.

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