Your Calculator Lies About Angles. Radians Are the Only One It Respects.

Start with the fact that trips up everyone: there is no single "angle." There are at least four units in active industrial use — degrees, radians, gradians, and revolutions — and they are not interchangeable by convention. They are convertible by exact constants, but only if you know which API is quietly demanding which one.

Radians Are the Only Honest Angle

A radian is defined by geometry, not by a committee. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. That single definition makes every circular formula clean: arc length = rθ, sector area = ½r²θ, and the angular velocity ω in radians per second is literally the tangential speed divided by the radius, with no constant of proportionality.

The cost of ignoring this is everywhere in engineering math. The identity d(sin θ)/dθ = cos θ is false if θ is in degrees — you get cos θ × π/180 instead. Every Taylor series, every Fourier coefficient, every control-loop derivative assumes radians. The discipline that saves you is boring but absolute: convert degrees (or grads) to radians at the boundary of any calculation, compute in radians, convert back only for display. Keep degrees inside a numeric integration and the simulation slowly, plausibly, spins apart.

For the pure conversion, the degrees to radians and radians to degrees tools on this site carry π to full precision so you never round the 0.0174532925… factor by hand.

Degrees: Babylon, Base-60, and the Bearing

Degrees come from the Babylonians, who liked 60 and its multiples — a circle of 360 is close to the days in a year and factors beautifully (2, 3, 4, 5, 6, 8, 9, 10, 12…). That divisibility is why we still slice a circle into 360 pieces nobody ever voted for.

Two notations live inside "degrees." Decimal degrees (38.8897°) are what a GPS spits out. Degrees-minutes-seconds (DMS) (38°53′23″) is what a pilot or surveyor reads, because base-60 lets you say "53 minutes" the way the Babylonians intended. Converting between them is just ×60 / ÷60 — but the off-by-one errors in DMS parsing (is it 0′–59′ or 1′–60′?) are a quiet source of coordinate bugs.

Navigation bearings are stated in degrees — clockwise from north, 000° to 359°. This is also why game engines and CSS lean on degrees for rotation: a human can picture "rotate 45 degrees" but not "rotate π/4 radians." The mismatch with the math libraries underneath is the entire subject of the failure section below.

Gradians: The French Revolution's Forgotten Unit

The gradian (called "gon" in ISO 80000-3, "grade" in older French texts) divides a right angle into 100 parts, so a full circle is 400 gradians. It was proposed during the French Revolution alongside the metric system, on the appealing logic that "90 degrees" should become the tidier "100 grads" and that all angles should be decimal.

It lost. Degrees kept the sky, the compass, and the classroom. But the gradian survived precisely where decimal angles are useful: European cadastral surveying and GIS. Many national land-registry systems store bearings in gon internally. When you import such a file into CAD, a game engine, or a CSS transform that expects degrees, you must multiply by 0.9. Miss that step and a 100-gon bearing becomes 100° — a 10° error that, at survey distances, misses a property corner by meters.

The practical conversion pair is gradians to degrees (multiply by 0.9) and its reverse. The factor is exact, but the detection of which unit your source file holds is the real hazard — verify the angular-unit column in the coordinate header before you trust any imported bearing.

Revolutions, RPM, and Turns

A full circle is also exactly one revolution: 360° = 2π rad = 400 gon = 1 turn. Motors and spindles are spec'd in RPM (revolutions per minute), but a control loop integrating angular position wants radians per second. The bridge is exact: 1 RPM = 2π/60 rad/s ≈ 0.10472 rad/s.

This is where angle conversion meets rotating machinery. A 3,000 RPM spindle is 314.16 rad/s. Feed that number to a controller expecting RPM and you have a 30× overspeed; feed RPM to one expecting rad/s and the axis crawls. The unit is the same physical quantity, converted by an exact constant — but the two halves of the system almost never agree on which constant to use, which is why datasheets print both.

Where Angle Conversion Breaks Things

The failures are rarely arithmetic — the constants are exact. They are expectation failures: one side of the interface assumes degrees, the other assumes radians (or grads, or encoder counts), and nobody wrote the unit down.

• CSS & SVG rotate(): expects degrees by default (rotate(45deg)). Pass a raw radian number and 1 radian renders as 57° — a spin that looks almost right. Some libraries accept rad only if you write the unit explicitly.
• Game engines (Unity, Unreal, Godot): rotation is usually degrees for transforms but radians for physics and math functions. Mixing them is the classic "my turret slowly rotates forever" bug.
• CNC rotary axes (G-code): angular moves are in degrees, but the servo controller and encoder may speak counts or radians internally. A post-processor that emits degrees into a radian-expecting controller destroys the part on the first rotary move.
• Robotics (Denavit–Hartenberg): joint angles are radians in the math, degrees in the teach pendant. A misread here is a robot through a wall.
• Astronomy: right ascension is often given in hours (24h = 360°), a fourth angle unit hiding inside a time word. Convert hours to degrees with ×15.

None of these is a hard problem. All of them have shipped as shipped bugs. The fix is the same one as every other unit on this site: declare the unit at every interface, convert once, carry the canonical unit (radians) internally.

The Numbers Themselves: Exact Angle Factors

Every factor below is exact. Degrees and radians are linked by π (a circle is 360° = 2π rad); degrees and grads by 400/360 = 10/9; radians and grads by 200/π.

Worked examples that are worth memorising: 180° = π rad, 90° = π/2 rad = 100 grad, 45° = π/4 rad = 50 grad, 30° = π/6 rad. If your conversion doesn't reduce to one of these at the familiar angles, the constant is wrong.

Which Converter to Reach For

This site carries three angle pairs. Each is exact both ways and carries π to full precision.

Degrees and radians

Use degrees to radians when feeding a degree reading into any math, physics, or control library; radians to degrees for the display direction.

Gradians

For European surveying and GIS data: gradians to degrees multiplies by 0.9. Always confirm the source column actually holds gon before you trust the result.

Questions We Get About Angle Units

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