Article — Degrees to Radians Converter
The Degrees to Radians Converter
A radian equals the angle that cuts an arc of length r on a circle of radius r. The degrees to radians formula is rad = deg × π/180, which gives 90° = π/2, 180° = π, and 360° = 2π. The radian is the SI unit of plane angle.
Most people first meet angles in degrees, where a full turn is 360. Mathematicians, physicists, and programmers prefer radians, where a full turn is 2π. The choice is not aesthetic — every derivative, integral, and arc-length formula in calculus gets shorter when you measure angles in radians.
What is a radian?
A radian is the central angle subtended by an arc whose length equals the radius of the circle. Take a circle of radius r, walk a distance r along its edge, and the angle from the centre is one radian. Because the full circumference is 2π r, the full angle is 2π radians — which is why 360° = 2π rad.
Calling the radian a "unit" understates its nature. It is the ratio of two lengths (arc divided by radius), so it has no dimension in the usual sense. The SI Brochure lists it as a dimensionless derived unit, which is why scientific writing often drops the rad symbol entirely and just writes sin(π/2) = 1.
The name radian was coined by James Thomson, an Irish mathematician, in the 1870s. Before that, mathematicians worked in radian measure without a dedicated word for it. The unit symbol rad was made official by the International Bureau of Weights and Measures in 1960.
The degrees to radians formula
The conversion is a single multiplication. To go from degrees to radians, multiply by π/180. To reverse it, multiply by 180/π.
deg → rad multiply by π/180 ≈ 0.01745rad → deg multiply by 180/π ≈ 57.296180° = π rad (anchor)360° = 2π rad (full turn)The factor π/180 is irrational. Any "exact" decimal you see (0.01745329 …) is truncated to whatever precision your tool allows. For most engineering and graphics work, six decimal places is plenty; for high-precision physics, you carry π itself as a symbol and substitute at the end.
Why radians matter for calculus and physics
The first identity that breaks down in degrees is the derivative of sine. In radians, the derivative of sin(x) is cos(x). In degrees, it is (π/180) × cos(x) — the same shape, but with a small constant attached forever. Every later identity inherits that extra factor, so trigonometric integrals become messy. Radians eliminate the bookkeeping.
Physics adopts radians for the same reason. The arc-length formula s = r θ assumes θ is in radians. So does angular velocity ω, where v = ω r. The Taylor series sin(x) ≈ x − x³/6 only works for x in radians. Stick with degrees and every equation needs an asterisk.
Common degrees to radians values
A handful of angles cover most trigonometry homework, most game-engine rotations, and most physics problems. Memorising them as fractions of π is the fastest way to read trig diagrams without reaching for a calculator.
- 30° = π/6 ≈ 0.5236 rad — appears in 30-60-90 triangles
- 45° = π/4 ≈ 0.7854 rad — the 45-45-90 triangle
- 60° = π/3 ≈ 1.0472 rad — equilateral triangle interior
- 90° = π/2 ≈ 1.5708 rad — quarter turn
- 180° = π ≈ 3.1416 rad — half turn
- 270° = 3π/2 ≈ 4.7124 rad — three-quarter turn
- 360° = 2π ≈ 6.2832 rad — full turn
Radians in code and graphics
Programming languages default to radians for trig. Python's math.sin, JavaScript's Math.sin, C's sin from math.h, Java's Math.sin — all expect radians. Excel does too, which is why =SIN(90) returns 0.893996663… instead of 1. To get sin(90°) in Excel you write =SIN(RADIANS(90)).
Game engines and 3D libraries follow the same convention. Unity exposes both radians and degrees in its Mathf class, but the internal math is radians. Unreal Engine uses radians for FMath::Sin and degrees only in the editor UI. OpenGL, WebGL, and DirectX rotation matrices are derived assuming radians.
If a trig calculation returns an unexpected result in code, check first whether you fed degrees into a radians-only function. The bug is usually a missing multiplication by π/180, not an algorithm error.
Degrees vs radians — when to use each
Degrees win when humans need to read or describe an angle. "45 degrees" sounds natural in a recipe, a carpentry instruction, or a weather forecast. Radians win for any math beyond basic geometry. The split lines up roughly like this:
- degrees — navigation, surveying, everyday geometry, instruction manuals, weather, compass bearings
- radians — calculus, physics, signal processing, robotics, graphics, programming, scientific calculators in RAD mode
- both — trigonometry textbooks (start with degrees, switch to radians by chapter 5)
A short history of the radian
Although the radian as a concept is older, the term itself was coined by James Thomson around 1873, building on earlier work by Roger Cotes. Cotes had already noted in 1714 that for trigonometric calculations the "natural" unit of angle is the one where 180° equals π. Once mathematicians settled on the radian, calculus textbooks started using it as the default within a generation.
The SI system formally accepted the radian as a supplementary unit in 1960 and reclassified it as a dimensionless derived unit in 1995. NIST and BIPM both note that the radian (rad) is the SI coherent derived unit of plane angle. The degree, written with a circle (°), is allowed alongside SI units but is not itself an SI unit.
Common degrees to radians mistakes
Most degrees to radians mistakes are mode-setting mistakes, not arithmetic mistakes. The conversion itself is trivial.
- DEG vs RAD mode — scientific calculators show DEG by default, languages and Excel default to RAD
- forgetting π/180 — writing Math.sin(45) when you want sin(45°) gives 0.851, not 0.707
- dropping the π — π/4 is not 1/4; it is ≈ 0.7854
- over-precision — π is irrational, so any decimal radian value is truncated; do not assume more accuracy than the formula allows
- mixing units in the same formula — every term in a trig identity must use the same angle unit
If sin or cos of a clean angle (90°, 180°, 270°) does not return the expected 1, 0, or −1, your unit setting is almost certainly the culprit. Convert to radians before passing values to a code library or spreadsheet, or change the calculator to DEG mode for textbook work.