Angle Converter
Convert between degrees, radians, gradians and revolutions.
Radians
3.141593
from 180 Degrees (°)
Degrees (°)
180
Gradians
200
Arcminutes
10,800
Revolutions
0.5
AI Breakdown & Smart Takeaway
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How the Angle Converter works
The GKCalculators Angle Converter instantly converts angle measurements between degrees, radians, gradians, and revolutions — making it an essential tool for students, engineers, programmers, and anyone working across disciplines where different angle systems collide.
At its core, this angle converter works by using a common reference unit to bridge all four supported systems. Every input value is first converted to a full revolution (one complete rotation), and then scaled into whatever target unit you need. Because a full revolution equals exactly 360°, 2π radians, 400 gradians, and 1 revolution, the converter can chain any pair of units together accurately without accumulating rounding errors from multi-step hops.
Degrees are the oldest and most intuitive unit, dividing a circle into 360 equal parts — a number chosen by ancient Babylonians for its rich divisibility. Radians are the mathematically natural choice: one radian is the angle subtended when the arc length equals the radius of the circle, and they make calculus and trigonometric identities far cleaner. Gradians (also called gon or grade) divide a right angle into 100 equal parts, giving a full circle exactly 400 gradians — a metrication-era unit still common in surveying and civil engineering across Europe.
Revolutions (or turns) are the least-used in academic contexts but extremely practical in mechanical engineering, motor specifications, and rotational physics, where quantities like RPM make the revolution the natural baseline. When you enter a value, the tool applies a direct proportional formula to convert to each output unit simultaneously, so you see all four representations at once. This is especially useful when debugging code, cross-referencing a physics textbook, or verifying a surveying instrument reading against a calculator output.
A common mistake is confusing radians with a 'bigger version' of degrees — they are dimensionless ratios, not arbitrary increments, and many programming languages (Python's math module, JavaScript's Math.sin(), C's trigonometric functions) expect angles exclusively in radians. Another frequent error is misidentifying gradians as degrees: 45° equals 50 gradians, not 45 gradians, so swapping the two in surveying software can introduce a subtle but significant geometric error. Using a reliable angle converter eliminates these silent mistakes before they propagate into calculations.
Formula
Degrees × (π / 180) = Radians | Degrees × (10 / 9) = Gradians | Degrees / 360 = Revolutions | Radians × (180 / π) = Degrees | Gradians × (9 / 10) = Degrees | Revolutions × 360 = Degrees
Pro tips
- If your programming language's trig functions return unexpected results, check first whether they expect radians — virtually all standard math libraries (Python, JavaScript, C, Java) use radians by default, not degrees.
- When working in surveying, remember that a 90° right angle is 100 gradians, not 90 — this makes gradian arithmetic very clean for grid-based land measurements and is worth keeping as a mental anchor.
- For quick mental estimation, remember that 1 radian ≈ 57.296°, so angles less than about 57° are less than 1 radian, which helps you sanity-check converter output on the fly.
- Convert to revolutions when dealing with rotational speed (RPM, angular velocity in cycles per second) before multiplying by 2π to get radians per second — this two-step approach reduces formula confusion in physics problems.
- Never round intermediate conversion results; let the tool carry full precision and only round your final displayed answer, since angle errors compound quickly in chained geometric calculations like those in 3D rendering or robot kinematics.
Key terms
- Degree (°)
- — A unit of angular measurement equal to 1/360 of a full rotation, originating from Babylonian base-60 mathematics and used as the everyday standard in navigation, geometry, and most non-scientific contexts.
- Radian (rad)
- — The SI unit of angle defined as the ratio of arc length to radius; a full circle contains exactly 2π radians, making it the natural unit for calculus, physics, and most programming math libraries.
- Gradian (gon / grad)
- — A metric angle unit that divides a right angle into 100 equal parts, giving 400 gradians per full circle; widely used in European surveying and civil engineering for its decimal-friendly alignment with right angles.
- Revolution (turn)
- — A unit representing one complete 360° rotation, commonly used in mechanical engineering and rotational dynamics where cycles or turns per minute are the natural way to describe angular motion.
- Arc Length
- — The distance along the curved edge of a circle subtended by an angle; the relationship arc length = radius × angle (in radians) is the geometric foundation from which the radian is defined.
- Dimensionless Ratio
- — A quantity with no physical units; radians are dimensionless because they represent a ratio of two lengths (arc to radius), which is why they can be dropped or inserted in equations without changing dimensional balance.