Degrees ⇄ Radians ⇄ Gradians Converter

Understanding the degrees, radians and gradians converter — concept and principle

Angles can be measured in several units. Degrees divide a full turn into 360 parts, radians measure the angle by arc length on a circle of radius 1, and gradians (also called gon) divide a turn into 400 parts. This converter takes an angle in degrees and gives radians, the radian value as a multiple of π, gradians, turns, and the degrees–minutes–seconds (DMS) form. It is used in trigonometry and calculus, programming (most math libraries expect radians), surveying, astronomy and map coordinates.

How it is calculated

• Radians = degrees × π ÷ 180.
• Multiple of π = degrees ÷ 180 (for example 90° = 0.5π).
• Gradians = degrees × 400 ÷ 360.
• Turns = degrees ÷ 360.
• DMS: the whole number is degrees; the fractional part × 60 gives minutes; the fractional part of that × 60 gives seconds.

The principle

A full turn is 360°, 2π rad, 400 grad and 1 turn, so every conversion is a fixed ratio. The radian is the natural unit: an angle of θ rad subtends an arc of length s = rθ on a circle of radius r. The circumference 2πr therefore corresponds to 2π rad. Because of this, calculus formulas such as d/dx sin x = cos x hold only in radians. The radian is the coherent SI unit of plane angle (equal to 1 m/m). Gradians were introduced with the metric system so that a right angle is exactly 100 grad. Minutes and seconds come from Babylonian base-60 counting: 1° = 60′ and 1′ = 60″.

Worked example

1. Default input: 180°.
2. Radians: 180 × π ÷ 180 = π = 3.141592654.
3. Multiple of π: 180 ÷ 180 = 1π.
4. Gradians: 180 × 400 ÷ 360 = 200.
5. Turns: 180 ÷ 360 = 0.5.
6. DMS: no fractional part, so 180° 0′ 0″.

How to read the result

Common values: 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 60° = π/3 ≈ 1.0472 rad, 90° = π/2 ≈ 1.5708 rad = 100 grad, 360° = 2π ≈ 6.2832 rad. One radian is about 57.2958°. In DMS, 12.345° is 12° 20′ 42″; in GPS coordinates one second of latitude is roughly 31 m on the ground.

Common mistakes

• Passing degrees to a programming function such as sin() that expects radians.
• Confusing decimal degrees with DMS: 12.5° is 12° 30′, not 12° 50′.
• Setting a calculator to "grad" mode by accident; it gives sin 90 = 0.988 instead of 1.

Frequently asked questions

Why do mathematicians prefer radians?

Because with radians the arc length is s = rθ and the derivative of sin x is cos x without extra factors of π/180.

How many degrees is one radian?

180 ÷ π ≈ 57.29577951°.

Where are gradians used?

Mainly in surveying and civil engineering in some European countries, where a right angle of exactly 100 gon simplifies calculations.

Sources: BIPM, The International System of Units (SI Brochure), 9th edition, 2019 — radian as the unit of plane angle · ISO 80000-3:2019, Quantities and units — Part 3: Space and time (plane angle)