Angle Converter

How to use this Calculator

The quick presets (30°, 45°, 60°, 90°, 180°, 270°, 360°) are the angles you’ll use most in trigonometry. Click any of them and every output updates immediately.

The 7 angle units, and when each one shows up

Most people use degrees their whole life and run into radians exactly once: the moment they paste an angle into Python, JavaScript, or a spreadsheet formula and get a completely wrong answer. That’s because code defaults to radians. Degrees are the human unit. Radians are the math unit.

Degrees to radians: the conversion everyone needs

This is the one that catches people most often. A 90-degree angle in a right triangle becomes π/2 radians in a formula. Same angle, completely different number.

The converter shows you the formula it used right below the result field. For 90° to radians, it displays: rad = ° ÷ 57.29578 → 90.000000° = 1.5707963 rad. That 57.29578 is 180/π, the conversion factor.

The unit circle and what sin, cos, tan actually mean

The converter shows sin, cos, and tan for whatever angle you enter. These aren’t just extra numbers. They’re coordinates.

That 6.12323e-17 for cos at 90° trips people up every time. It’s not a bug. It’s floating-point arithmetic. The true value is 0, but computers can’t represent π/2 exactly in binary, so the result is an astronomically small number that rounds to zero for any practical purpose.

Key angles you’ll use constantly

At 45°, sin and cos are identical (both √2/2 ≈ 0.707). That makes 45° the perfectly balanced angle, the one where horizontal and vertical components are equal. It’s why 45° shows up so often in physics problems involving projectiles.

Conversion formulas for all 7 units

Gradians: the metric angle unit that never quite caught on

The gradian (also called gon) divides a full circle into 400 equal parts. A right angle is exactly 100 gradians. The idea was to make angles decimal-friendly the same way the metric system made lengths decimal-friendly.

France introduced gradians during the same push that gave us kilometers and kilograms. The metric system stuck. Gradians didn't, except in civil engineering and land surveying, where a right angle being exactly 100 units actually simplifies the calculations. Surveyors in continental Europe still use them. Everyone else uses degrees.

The practical advantage: slope percentages in civil engineering are calculated in gradians naturally. A 1% slope is 0.573°, which isn’t memorable. In gradians, slope calculations map more cleanly to the numbers surveyors work with daily.

Arcminutes and arcseconds: where angles get very small

An arcminute is 1/60 of a degree. An arcsecond is 1/60 of an arcminute, or 1/3600 of a degree. These units exist because degrees are too coarse for precise astronomical and navigational measurements.

When astronomers say a star is 1 arcsecond away from another, they mean the angular separation in the sky is 1/3600 of a degree. The Hubble Space Telescope can resolve objects 0.05 arcseconds apart. For context, that’s like reading a newspaper from 2 kilometers away.

GPS coordinates use degrees, arcminutes, and arcseconds written as DMS notation. 48° 51’ 30” N is Paris, latitude expressed in degrees, arcminutes, and arcseconds. The converter can work with any of those values individually.

Milliradians: why snipers use this instead of degrees

A milliradian (mrad) is 1/1000 of a radian. At 1,000 meters, 1 mrad corresponds to exactly 1 meter of lateral movement. That relationship stays proportional at any distance.

Artillery and naval gunfire use mils (a related but slightly different unit, 1/6400 of a full circle instead of 1/2000π) for the same reason. Angular units that produce clean arithmetic at combat distances are operationally useful in a way that degrees are not.

Turns: the rotation unit making a comeback

A turn is one full rotation. 360° = 1 turn. It’s the most intuitive unit for mechanical systems and animations.

CSS actually uses turns as a unit in the rotate() function. transform: rotate(0.25turn) rotates an element 90°. rotate(0.5turn) is 180°. Engineers specifying motor rotations sometimes use turns per minute rather than degrees per second because it maps directly to RPM.

The equivalent angles output: reading all 7 at once

The “All Equivalent Angles” section of the calculator shows your input converted to every unit simultaneously. For 90°, that table shows:

The 100 gradians result for a right angle is why surveyors like gradians. Every right angle in a survey grid is a clean 100 grad. Trigonometric calculations on rectangular plots become slightly cleaner because the right angle is a round number.

Where each field actually uses these units

The one conversion mistake that breaks everything

If you’ve ever written Math.sin(45) in JavaScript and gotten 0.8509 instead of 0.7071, this is why.

Math.sin(45) interprets 45 as 45 radians, not 45 degrees. 45 radians is about 2578 degrees, an angle that’s gone around the unit circle more than 7 times. The sin of that is 0.8509.

The converter’s formula display (the line that shows rad = ° ÷ 57.29578 → 90.000000° = 1.5707963 rad) is specifically useful for this. Copy the output radian value. Paste it into your function. Done.

Negative angles and angles over 360°

The converter handles both without complaining. A negative angle means clockwise rotation in standard mathematical convention. -90° is the same point on the unit circle as 270°, just described as rotating backward from zero.

Angles over 360° are valid too. 450° is one full rotation plus 90°, which lands at the same spot as 90°. The sin, cos, and tan values for 450° are identical to those for 90°. The converter shows you the equivalent angles panel with the canonical form (90°) alongside your input.