Degrees, Radians, Gradians and Turns: How to Convert Any Angle

An angle is one idea, but it shows up wearing four different costumes. Your geometry textbook writes 90 degrees. Your code expects 1.5708. A European survey plan prints 100 gon. A physicist might just say a quarter turn. They all describe the same corner, yet plug the wrong number into the wrong place and your result quietly falls apart. This guide walks through each unit, where it actually gets used, and the exact factors that move you between them.

The four units, side by side

Start with the one fact that ties everything together. A full circle can be written four ways at once:

360° = 2π rad = 400 grad = 1 turn

Each unit just slices that same loop into a different number of pieces.

Degrees cut the circle into 360 parts. The number 360 is ancient, inherited from Babylonian base-60 counting, and it survives because it divides cleanly: 360 splits into halves, thirds, quarters, fifths, sixths, eighths, ninths and tenths without fractions. That makes degrees friendly for everyday geometry, navigation and construction.
Radians measure an angle by the arc it carves out. One radian is the angle whose arc length equals the circle's radius. Wrap the radius around the rim and you fit exactly 2π of them, so a full turn is 2π radians, roughly 6.2832. Radians look messy to humans but they are the natural language of calculus.
Gradians (also written gon or grad) split the circle into 400 parts, which makes a right angle exactly 100. Surveyors on the European continent like the round numbers, especially for slope and percentage-grade work.
Turns count whole revolutions. One turn is the entire circle, half a turn is a straight angle, a quarter turn is a right angle. It is the most intuitive unit and increasingly common in animation and game engines, where "rotate by 0.25" reads cleaner than "rotate by 90."

Why math and code live in radians

Here is the part that trips up most beginners. If you have ever written Math.sin(90) in JavaScript or math.sin(90) in Python and gotten a bizarre answer, this is why: those functions read their input as radians, not degrees. Math.sin(90) computes the sine of 90 radians, which is about 0.894, not the 1 you wanted for a right angle.

Radians are not a stylistic choice in mathematics; they fall out of the calculus itself. The derivative of sin(x) is cos(x) only when x is in radians. The small-angle approximation sin(x) ≈ x holds only in radians. Power series, Fourier transforms, every trig identity in a physics paper assumes radians underneath. So programming languages standardize on them, and you convert at the boundary: take the user's degrees, turn them into radians, do the math, and convert back for display. If you are checking trig values by hand, the trigonometry calculator lets you set the mode explicitly so you never guess which unit a function expects.

Degrees, by contrast, win wherever a human reads the number directly. Compass bearings, a protractor in a classroom, the tilt of a solar panel, the angle of a staircase. Nobody describes a steep roof as "0.6 radians." The unit you pick should match the audience: degrees for people, radians for equations.

The conversion factors you actually need

There are only two factors worth memorizing, and everything else falls out of the full-circle identity above.

To go from degrees to radians, multiply by π and divide by 180:

radians = degrees × (π / 180)

To go the other way, flip the fraction:

degrees = radians × (180 / π)

That 180/π works out to about 57.29578, so one radian is roughly 57.3 degrees. For the other units, lean on the full-circle equivalence. Degrees to gradians is a clean × 10/9 (because 400/360 = 10/9), so 90° becomes 100 gon. Degrees to turns is simply ÷ 360. And one degree breaks down further into 60 arcminutes, each of which splits into 60 arcseconds, the same way a clock face does.

A worked example

Let me take the angle that confuses people most, the right angle, and run it through the degrees-to-radians factor.

Start with 90°. Multiply by π/180:

90 × (π / 180) = 90π / 180 = π / 2 ≈ 1.5708 radians

So 90 degrees is exactly π/2 radians, which evaluates to about 1.5708. That is the number your code needs. Do the same with 45°:

45 × (π / 180) = π / 4 ≈ 0.7854 radians

Notice the pattern: 45° is half of 90°, and 0.7854 is half of 1.5708. The factor is linear, so once you have one anchor the rest scale directly. A few worth keeping in your head: 180° is π (about 3.1416), 360° is 2π (about 6.2832), and 30° is π/6 (about 0.5236).

When I first started writing canvas animations, I lost an entire evening to a rotation that spun far too fast. I had read "90 degrees per frame" off a design note and passed 90 straight into a rotate call that expected radians. Ninety radians is more than fourteen full turns, so every frame the sprite whipped around fifteen times. The fix was one line, 90 * Math.PI / 180, but the lesson stuck: always convert at the edge, and never assume a library shares your unit.

Picking the right unit for the job

There is no single best angle unit. Each one earns its place:

Use degrees for anything a person reads or sets by hand: navigation, geometry homework, camera tilts, construction, weather (wind direction).
Use radians for any real computation: trig functions in code, physics, calculus, signal processing, robotics joint angles.
Use gradians if you work with European survey data or want a right angle to equal a round 100 for percentage-grade math.
Use turns for rotational quantities and animation, where whole revolutions are the natural mental model.

The trap is mixing them silently. A degree value that looks small (100°) is a large number in radians (1.745), and 100 gon is not 100 degrees, it is 90. Always lock in which unit you are reading before you trust the number.

When you need to move between any of these without doing arithmetic in your head, the angle converter takes one value and shows degrees, radians, gradians, turns, arcminutes, arcseconds and NATO mils all at once, with degrees as the hub so every result stays consistent. Type 90, and you see π/2, 100 gon and a quarter turn lined up where you can sanity-check the whole set at a glance.

Angles are simple once the units stop fighting you. Memorize the full circle, keep the π/180 factor handy, and convert at the boundary between what humans read and what machines compute.

Made by Toolora · Updated 2026-06-13