What the Angle Converter Does and Who It's For

This angle converter changes a single angle measurement between four common units: degrees, radians, gradians, and turns. Type a value in any one field and the others update automatically, so you never have to juggle conversion factors by hand.

It's built for students working through trigonometry and calculus, engineers and surveyors who mix degrees with gradians, and programmers who need radians because functions like sin() and cos() in most languages expect radian input. Whether you're converting degrees to radians for a physics problem or checking a CAD bearing, the tool gives an exact answer in seconds.

How It Works: The Conversion Formulas

Every conversion runs through one base unit: the radian. A full circle equals 1 turn = 360 degrees = 2π radians = 400 gradians. From those equalities, all the factors follow.

The core relationships are written out below. π (pi) is approximately 3.14159265.

Degrees to radians: radians = degrees × (π / 180)
Radians to degrees: degrees = radians × (180 / π)
Degrees to gradians: gradians = degrees × (400 / 360) = degrees × (10 / 9)
Degrees to turns: turns = degrees / 360
Radians to turns: turns = radians / (2π)
Gradians to degrees: degrees = gradians × (9 / 10)

Worked Example: Converting 90 Degrees

Suppose you want to convert a right angle of 90 degrees into the other three units.

Radians = 90 × (π / 180) = π / 2 ≈ 1.5708 rad. Gradians = 90 × (10 / 9) = 100 grad. Turns = 90 / 360 = 0.25 turn. So a 90-degree angle is exactly a quarter turn, 100 gradians, and π/2 radians.

Reversing one of these confirms the math: 1.5708 rad × (180 / π) = 1.5708 × 57.2958 ≈ 90 degrees. The factor 180/π ≈ 57.2958 is the number of degrees in one radian, worth memorizing.

Where These Units Are Used

Radians are the natural unit in mathematics, physics, and programming because they simplify calculus (the derivative of sin x is cos x only when x is in radians) and feed directly into trig functions in Python, JavaScript, C, and most scientific calculators set to RAD mode.

Degrees dominate everyday geometry, navigation, and construction. Gradians (also called gons) appear in European surveying and some engineering, where a right angle of 100 grad makes percentage-style slope math easier. Turns (revolutions) are handy for rotational motion, gear ratios, and angular displacement.

Tips and Common Mistakes

The single most frequent error is leaving a calculator or code in the wrong mode. If your calculator is in DEG mode but the formula expects radians, sin, cos, and tan will return wrong values. Always confirm the mode before trusting trig output.

Don't approximate π as 3.14 for precise work; use the full value to avoid rounding drift in long calculations.
Remember radians are dimensionless (a ratio of arc length to radius), so they often appear without a unit symbol.
Negative and angles over 360 degrees convert the same way; subtract multiples of 360 only if you need a value within one rotation.
Gradians use 400 per circle, not 360 or 100, a common slip when first learning the unit.
Keep extra decimal places during intermediate steps and round only the final answer.