Radian to Grad (Gon)
rad
grad
Conversion History
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Quick Reference Table (Radian to Grad (Gon))
| Radian (rad) | Grad (Gon) (grad) |
|---|---|
| 0.5 | 31.83098861837906956991 |
| 1 | 63.66197723675813913982 |
| 1.5708 | 100.00023384349968496082 |
| 3.1416 | 200.00046768699936992164 |
| 4.7124 | 300.00070153049905488248 |
| 6.2832 | 400.0009353739987398433 |
About Radian (rad)
The radian (rad) is the SI unit of plane angle, defined as the angle subtended at the center of a circle by an arc whose length equals the circle's radius. Because it is defined as a ratio of two lengths, the radian is dimensionless. A full circle spans exactly 2π radians (≈6.2832 rad). Radians are the natural unit in calculus, physics, and engineering: trigonometric functions in mathematics and most programming languages use radians by default, and angular frequency in mechanics and electronics (ω = 2πf) is expressed in radians per second.
One radian is approximately 57.3°. In physics, a pendulum's small-angle approximation (sin θ ≈ θ) is valid only when θ is in radians and small.
Etymology: The term "radian" was coined around 1873 by Irish mathematician James Thomson. The concept emerged naturally from defining angles via the ratio of arc length to radius — a ratio used implicitly in trigonometry since antiquity.
About Grad (Gon) (grad)
The grad (also called gon or grade, symbol: grad or g) divides a full circle into 400 equal parts, so a right angle is exactly 100 grad. It was introduced during the French metric reform of the late 18th century to create a decimal-friendly angular system compatible with metric measurements. The grad persists in civil engineering, land surveying, and mining in continental Europe, particularly in France, Germany, and Scandinavia. Most scientific calculators include a GRAD mode alongside DEG and RAD.
A slope of 1 grad in road engineering is a 1 gon incline from horizontal — used in surveying instruments and tachymeters across Europe.
Etymology: From the French "grade", introduced around 1793 as part of the revolutionary metric system. The 400-division was chosen so that a right angle equals exactly 100 grad, aligning with decimal arithmetic.
Radian – Frequently Asked Questions
Why do mathematicians and physicists prefer radians over degrees?
Radians make calculus work cleanly. The derivative of sin(x) is cos(x) — but only if x is in radians. In degrees, the derivative picks up an ugly π/180 factor that contaminates every formula. Angular frequency (ω = 2πf), rotational kinetic energy, wave equations, and Euler's formula (e^(iπ) = −1) all assume radians. Degrees would litter physics with conversion constants the way imperial units litter engineering. Radians aren't a preference — they're the unit that makes the math not lie to you.
What does one radian actually look like?
Imagine wrapping the radius of a circle along its curved edge — the angle that arc subtends at the center is one radian. It works out to about 57.3°, which is a little less than the angle of an equilateral triangle's corner (60°). A pizza slice cut at one radian would be a generous but not absurd portion — wider than a sixth of the pie but narrower than a quarter. It looks unremarkable, which is ironic given how fundamental it is.
Why do programming languages use radians for trigonometric functions?
Every major language — C, Python, JavaScript, Java, Rust — uses radians in Math.sin(), Math.cos(), and related functions because the underlying floating-point hardware and Taylor series expansions assume radian input. The Taylor expansion of sin(x) is x − x³/3! + x⁵/5! − … and only converges correctly when x is in radians. Feeding in degrees without converting first is one of the most common bugs in student code and game physics.
How do you quickly convert between radians and degrees in your head?
Memorise that π radians = 180°. From there: multiply radians by 180/π (roughly 57.3) to get degrees, or multiply degrees by π/180 to get radians. The common angles are worth memorising outright — π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360°. If you forget, just remember that 1 radian ≈ 57° and estimate from there.
Is a radian truly dimensionless or does it have units?
Officially dimensionless. The radian is defined as arc length divided by radius — meters over meters — so the units cancel. The SI classifies it as a "supplementary unit" turned "derived unit with the special name radian." This dimensionlessness causes genuine headaches: torque (N·m) and energy (J = N·m) have identical SI dimensions, and only the implicit "per radian" distinguishes them. Some physicists argue the radian should be treated as a base unit to avoid exactly this confusion.
Grad (Gon) – Frequently Asked Questions
Why did the French Revolution create the gradian when degrees already worked fine?
The revolutionaries wanted to decimalize everything — length (meter), mass (kilogram), time (decimal hours), and angle. The gradian divided a right angle into exactly 100 parts, making it compatible with the new metric system and decimal arithmetic. A slope of 1% grade corresponds neatly to gradian-based calculations. Decimal time flopped within a year, the Republican Calendar lasted 12 years, but the gradian quietly survived in surveying because it genuinely simplifies land measurement calculations.
What is the difference between a gradian, a gon, and a grade?
They're all the same unit — 1/400 of a circle. "Gradian" is the international English term, "gon" is preferred in German-speaking countries and ISO standards, and "grade" is the original French name. The symbol varies too: grad, gon, or superscript g. This naming mess is partly why the unit never gained traction outside continental Europe — nobody could agree on what to call it.
Where are gradians still actually used today?
Primarily in civil engineering and land surveying in France, Germany, Switzerland, and Scandinavia. Total stations (electronic surveying instruments) from Leica and other European manufacturers default to gon. Mining engineers in Germany use gon for underground surveys. French national mapping uses grades for geodetic calculations. If you buy a Leica total station in Europe, you may need to switch it from gon to degrees before using it elsewhere.
Why does every scientific calculator have a GRAD mode that almost nobody uses?
Calculator manufacturers include DEG, RAD, and GRAD modes because international standards (particularly IEC 60747) require it, and European civil engineering exams expect students to work in gradians. The mode exists for a real user base — it's just a user base concentrated in specific countries and professions. The most common calculator accident in the world is probably getting wrong trig answers because the calculator was accidentally left in GRAD mode after someone else used it.
How do you convert between gradians and degrees?
Multiply gradians by 0.9 to get degrees, or multiply degrees by 10/9 to get gradians. A right angle is 100 grad = 90°. The conversion factor is 9/10 because 400/360 = 10/9. This means 50 grad = 45°, 200 grad = 180°, and 300 grad = 270°. The relationship is simple enough to do in your head, which is one of the few nice things about having two competing angular systems.