Verification: factors follow standard unit definitions; round for display only.

Degrees to radians

Convert ° to rad when a drawing, compass bearing, or classroom problem lists degrees but your trigonometry library, physics equation, or navigation algorithm expects radians.

A full rotation is 360° and 2π rad, so 180° = π rad. Degrees divide the circle into familiar steps; radians measure arc length along a unit circle, which is why sin, cos, and most calculus formulas use radian arguments.

rad = ° × (π/180)  ·  equivalently   rad = ° ÷ 57.2958

This site's angle catalog stores 1 radian = 57.2958 degrees, so one degree equals 0.0174533 radians—the same value as π/180. Multiply degrees by that factor (or divide by 57.2958); round only at the end for labels or UI.

Step-by-step conversions (worked examples)

90° (quarter turn, right angle):

1. 90 × (π/180) = 90 ÷ 2 = π/2 rad
2. Numeric: 1.5708 rad

180° (straight angle, half turn):

1. 180 × (π/180) = π rad
2. Numeric: 3.14159 rad

45° (common in navigation and isosceles right triangles):

1. 45 × (π/180) = π/4 rad
2. Numeric: 0.785398 rad

Degrees to radians conversion chart

Where degrees → radians comes up

• Trigonometry: Spreadsheets and languages compute sin(θ) in radians—convert a 30° input before calling the function.
• Engineering: Phase angles in AC circuits, shaft rotation in kinematics, and antenna beamwidth specs often arrive in degrees but feed radian-based models.
• Programming: JavaScript uses deg * (Math.PI / 180); Python offers math.radians(deg). Forgetting this step is a top cause of wrong trig plots.
• Navigation: Compass courses and chart bearings are usually in degrees; great-circle and dead-reckoning formulas in libraries often require radians internally.

Radians to degrees

Convert rad to ° when a solver, sensor log, or code output reports radians but you need degrees for drawings, compass roses, or field notes.

Invert the forward relationship: multiply radians by 180/π, or use the catalog factor 57.2958 (degrees per radian).

° = rad × (180/π)  ·  equivalently   ° = rad × 57.2958

Step-by-step conversion (worked example)

Convert 1 rad to degrees:

1. 1 × (180/π) = 57.2958°
2. Sanity check: one radian is the angle whose arc on a unit circle equals one radius length.

Convert π/2 rad back to degrees: (π/2) × (180/π) = 90°—recovering the right-angle anchor from above.

Radians to degrees conversion chart

Reverse conversion is essential when exporting simulation results to CAD title blocks or when comparing a robot joint encoder (radians) against a mechanical drawing (degrees).

Formulas compared, programming snippets, and related tools

Both directions use exact definitional factors tied to π. Keep unit symbols explicit, avoid double rounding, and verify with anchor angles.

Degrees vs radians at a glance

Programming: degrees ↔ radians

• JavaScript / TypeScript: const rad = deg * (Math.PI / 180); reverse with deg = rad * (180 / Math.PI).
• Python: math.radians(deg) and math.degrees(rad).
• Excel: RADIANS(deg) and DEGREES(rad).

Common mistakes to avoid

• Passing degrees to sin/cos without converting—30° is not 30 rad.
• Using 3.14 instead of π in repeated conversions—fine for rough sketches, not for chained engineering calculations.
• Mixing gradians or arc minutes with degree inputs—confirm the symbol on the source document first.
• Rounding too early—carry full precision through multi-step work; round once at display.

Exactness and round-trip verification

The relationship π rad = 180° is exact by definition. Converting 45° → rad → 45° should recover the original within floating-point limits. Test anchors: 90° = π/2 rad, 180° = π rad, 1 rad = 57.2958°.

Related angle converters

For the inverse route, see radians to degrees. For subdivisions of a degree, use degree to arc minute, arc minute to degree, and arc second to degree. For the gradian (gon) system used in some surveying tools, see degree to gradian and gradian to degree.