📐 Degrees ⇄ Radians Converter
45° = π/4 — not 0.7853981634. Exact π-fraction conversion both ways, plus a degrees-minutes-seconds panel for GPS-style coordinates.
135° = 3π/4 rad ≈ 2.356194
Working
Multiply by π/180 and reduce the fraction: 135° × π/180 = 135π/180 = 3π/4. The exact π form is what trigonometry and calculus expect — sin(π/4), not sin(0.7854).
Exact π fractions — 45° is π/4, never 0.7853981634. Runs locally.
How the degrees ⇄ radians converter works
Degrees to radians multiplies by π/180 and reduces the fraction exactly: 45° → 45π/180 → π/4, and 137° → 137π/180 (already in lowest terms — not every angle is pretty, and the exact form says so honestly). Radians to degrees accepts the coefficient of π (enter 5/6 for 5π/6) and multiplies by 180 exactly. The DMS panel converts degrees-minutes-seconds to exact decimal degrees via 1′ = 1/60° and 1″ = 1/3600°, keeping the result as a fraction before showing the decimal. Common angles (30°, 45°, 60°, 90°…) are one click away with their π forms.
The exact π form is not pedantry — it's what the next step of the mathematics needs. Trigonometric exact values are defined at π/6, π/4, π/3; calculus formulas assume radians; and sin(45°) computed as sin(0.7853981634) carries a rounding error into everything downstream. Decimal radians are shown alongside for numerical work, clearly marked as the approximation.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π/180 and simplify: 60° = 60π/180 = π/3. The converter reduces the fraction exactly and shows the step — the decimal (≈1.0472) is the approximation, not the answer.
Why are radians the "natural" angle unit?
A radian is the angle whose arc equals the radius, which makes the calculus clean: sin′ = cos only in radians, and arc length is simply rθ. That's why every formula past trigonometry assumes radians.
How do I enter a radian value like 5π/6?
Type the coefficient of π — 5/6 — in the radians tab; the tool returns 150° exactly. Decimal radian inputs without π are rarely what a course means, and the exact path avoids that trap.
What is DMS notation?
Degrees-minutes-seconds: 30° 15′ 50″ means 30 + 15/60 + 50/3600 degrees — the format of GPS coordinates, astronomy and surveying. The tool converts it to an exact fraction of a degree, then decimal.
Is the conversion exact?
The π-fraction and DMS arithmetic are exact rational math; only the optional decimal display rounds. Runs locally.