Angles can be measured in three common units — degrees, radians, and gradians — and each shows up in a different context: degrees in everyday navigation and construction, radians in calculus and physics, and gradians in surveying. This calculator converts freely between all three and highlights when a radian value lands on a clean fraction of π, which matters for trigonometry, calculus, and unit-circle work.
Why radians matter in math and science
Degrees are convenient for everyday use, but radians are the natural unit for calculus and physics because they make derivative and integral formulas for trigonometric functions clean — for example, the derivative of sin(x) is cos(x) only when x is in radians. A radian is defined geometrically: it's the angle where the arc length equals the radius, which is why a full circle (circumference 2πr) spans exactly 2π radians regardless of the circle's size.
Reading the exact π form
Many angles that come up constantly in trigonometry — 30°, 45°, 60°, 90°, and their multiples — convert to clean fractions of π in radians: π/6, π/4, π/3, and π/2 respectively. Recognizing these exact forms (rather than their decimal approximations) makes it much easier to work with trig identities, unit-circle diagrams, and exact-value problems common in precalculus and calculus courses. This calculator's π Form calculation view checks denominators up to twelfths, which covers every standard angle taught in a typical trigonometry course.
When gradians come up
Gradians divide the circle into 400 parts instead of 360, so a right angle is a round 100 gradians and a full circle is 400. They're rarely used outside of surveying, some artillery and geodesy applications, and a handful of European engineering contexts, but calculators and some software still support them as a third angle unit. If you're not working in one of those fields, you can safely ignore gradians and stick to degrees and radians.
Use finite values with magnitude at most 1e50. Expressions support pi, sqrt(), fractions and arithmetic. Angles are limited to 1e12 degrees. Units are mathematical; geometry alone does not establish physical safety. Read the linked values, named formulas, diagram coordinate table and any branch or parameter comparison.