Angle converter

Angles can be measured in different units; the most common is the degree (°), but trigonometry and physics use the radian.

Equivalences

• A full turn = 360° = 2π radians = 400 gradians
• 180° = π rad  ·  90° = π/2 rad
• 1 rad ≈ 57.2958°

Degree ↔ radian

rad = degrees × π / 180  and  degrees = rad × 180 / π

Step-by-step example

90° in radians: 90 × π/180 = π/2 ≈ 1.571 rad.

Why the radian exists

The degree is a historical convention — 360 probably because it divides evenly by so many numbers and approximates the days in a year. The radian is not a convention: 1 radian is the angle at which the arc equals the radius in length. That is what makes formulas simpler: arc length becomes s = r × θ with no constant at all, and the derivatives of sine and cosine are only clean in radians. It is why scientific calculators have a RAD mode.

Gradians and turns

The gradian (or gon) divides a full turn into 400 parts so that a right angle is exactly 100 — an attempt to decimalise angle alongside the metric system. It survives in surveying in some countries. The turn is the most direct unit of all: 1 turn = 360° = 2π rad, and it appears in rotation and frequency.

Watch the calculator mode

The most common trigonometry error is computing a sine in radians while believing you are in degrees. sin(30) is 0.5 in degrees and −0.988 in radians — two plausible, completely different results, and nothing in the answer signals which mode produced it.