Trigonometry formulas

Degrees and radians

Convert an angle between degrees, radians, gons and turns, reduce it to one full turn, and read the exact multiple of π.

Formula

α [rad] = α [°] \times \frac{π}{180}

One full turn equals 360° and 2π rad, so a single degree is π/180 radians. The radian is the coherent derived SI unit of plane angle and has dimension one.

Degrees and radians calculator

Convert to

Given

Angleα

Reduce to the range 0°–360° (0–2π)Removes whole turns, so 450° becomes 90° and −90° becomes 270°.

The result updates automatically. You may use a decimal point.

Result

1.047198 rad

Exact form: π/3 rad

The same angle in other units

degrees: 60 °
radians: 1.047198 rad
gons: 66.666667 gon
turns: 0.166667 tr

Substitution and calculation

α[rad] = α[°] × π / 180

= 1.047198 rad

Symbols and units

Symbol	Quantity	SI	Supported units
α	anglePlane angle written in the unit you start from: degrees, radians, gons or turns.	rad	rad, °, gon, tr
β	converted angleThe same angle written in the target unit; only the notation changes, not the angle itself.	rad	rad, °, gon, tr
α₀	normalised angleAngle reduced to a single full turn: 0 ≤ α₀ < 360°, that is 0 ≤ α₀ < 2π rad.	rad	rad, °, gon, tr
k	whole turnsNumber of complete turns removed during normalisation; it is a count, so it has no unit.	1	1

Formula rearrangements

Degrees to radians

α[rad] = α[°] × π / 180

Radians to degrees

α[°] = α[rad] × 180 / π

Degrees and gons

α[gon] = α[°] × 10 / 9

Angle as a fraction of a turn

k = α[°] / 360 = α[rad] / (2π)

Reduce to one full turn

α₀ = α − 360° × ⌊α / 360°⌋

Worked examples

Sixty degrees in radians

Express the angle 60° in radians and give the exact multiple of π.

1Step 1: Use α[rad] = α[°] × π / 180.
2Step 2: Simplify the fraction: 60 / 180 = 1/3.
3Step 3: Multiply by π to obtain π/3 rad, approximately 1.0472 rad.

Result: π/3 rad ≈ 1.0472 rad

Angle larger than a full turn

Reduce 450° to the range 0–360° and write the result in radians.

1Step 1: Count the whole turns: 450 / 360 = 1.25, so one full turn is removed.
2Step 2: Subtract it: 450° − 360° = 90°, which is the same direction on the unit circle.
3Step 3: Convert the normalised angle: 90° × π / 180 = π/2 rad, approximately 1.5708 rad.

Result: 90° = π/2 rad ≈ 1.5708 rad

Sources and review

The International System of Units (SI), 9th edition
Bureau International des Poids et Mesures (BIPM) · Reviewed 29 August 2026
Definition of the radian as the coherent derived SI unit of plane angle with dimension one.
Guide for the Use of the International System of Units (SI), Special Publication 811
National Institute of Standards and Technology (NIST) · Reviewed 29 August 2026
Rules for units accepted for use with the SI, including the degree, and for writing unit symbols.
Algebra and Trigonometry 2e, section 7.1: Angles
OpenStax · Reviewed 29 August 2026
Degree and radian measure, coterminal angles, and worked conversions.

Assumptions and scope

One full turn is 360° = 2π rad = 400 gon = 1 turn; every conversion on this page follows from that single definition.
The radian is the coherent derived SI unit of plane angle and has dimension one, so the symbol rad may be written or omitted; the degree and the gon are units accepted for use with the SI, not SI units.
The radian measures a plane angle. It must not be mixed with the steradian, which is the SI unit of solid angle.
Normalisation reduces an angle to the half-open range 0 ≤ α₀ < 360°, so 360° becomes 0° and −90° becomes 270°.
The result is shown as an exact multiple of π only when the angle is a rational multiple of π; otherwise the decimal value is used.

Common mistakes

Multiplying by 180/π when converting degrees to radians; that factor converts radians to degrees.
Leaving a calculator in degree mode while the argument of a trigonometric function is given in radians.
Confusing the radian, which measures a plane angle, with the steradian, which measures a solid angle.
Treating one gon as one degree; a gon equals 0.9°, because a full turn has 400 gons.
Normalising −90° to 90° instead of 270° by dropping the sign instead of adding a full turn.