Angle Converter

Degrees, radians, and gradians

An angle measures how far something turns. Three scales appear often in maths, surveying, and engineering:

• Degrees (°) — a full circle is 360°. Everyday geometry, navigation bearings, and most school maths use degrees.
• Radians (rad) — a full circle is 2π radians. Calculus, physics, and programming trigonometry libraries almost always expect radians.
• Gradians (gon / grad) — a full circle is 400 gradians. Still used in some surveying and European civil-engineering contexts because a right angle is a clean 100 gon.

Conversion factors

All three describe the same turn. This page keeps them linked through one internal angle so editing any field updates the others exactly:

• radians = degrees × π / 180
• degrees = radians × 180 / π
• gradians = degrees × 10 / 9
• degrees = gradians × 9 / 10

Worked example

A right angle is 90°, which is π/2 ≈ 1.5708 rad and 100 gradians. A 45° mitre is about 0.7854 rad or 50 gon. If a programming API needs Math.cos input, convert degrees to radians first — feeding 90 into a radian cosine returns almost zero, not the cosine of a right angle.

When each unit helps

Stay in degrees for DIY, maps, and diagram labels. Switch to radians when checking identities like small-angle approximations (sin θ ≈ θ for θ in radians) or when debugging trig code. Use gradians only when a plan or instrument is already marked in gon so you avoid a double conversion through degrees.

Common mistakes

• Leaving a calculator or language runtime in the wrong angle mode (DEG vs RAD).
• Wrapping angles inconsistently (for example mixing −180…180 with 0…360 bearings).
• Confusing gradians with gradients as in slope percentage — different concepts.

FAQs

Why do computers prefer radians?

Power-series definitions of sin and cos are simplest when the angle is measured in radians.

Are gradians the same as grades?

In surveying, “gon” and “grad” usually mean the 400-per-circle unit shown here, not a percentage slope.

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