Convert degrees to radians
radians = degrees × π / 180
The degree cuts the turn into 360 parts, the radian measures it by arc length: one turn is 2π radians, and the exact conversion factor is π/180. This page converts both ways, as well as to gradians, turns, arcminutes and arcseconds, with the factor shown under the result. It is also the conversion hiding behind the most widespread calculator mistake: a sine computed in the wrong mode.
1 Degree (°) = 0.01745329252 Radian (rad)
Raw value: 0.0174533
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasradians = degrees × π / 180
radians = degrees × π / 180
degrees = radians × 180 / π
1 turn = 360° = 2π rad = 400 gon
1° = 60′ = 3600″
The π/180 factor is exact by definition: the degree is defined as 1/360 of a turn, the radian as the angle whose arc equals the radius, and the full turn is 2π radians. Only the decimal writing of π is approximate; the engine uses the double-precision value, the same as any scientific calculator.
- 30°
- · π/6 ≈ 0.5236 rad; the sine is 1/2.
- 45°
- · π/4 ≈ 0.7854 rad; sine and cosine equal, √2/2.
- 60°
- · π/3 ≈ 1.0472 rad; the cosine is 1/2.
- 90°
- · π/2 ≈ 1.5708 rad: the right angle.
- 180°
- · π ≈ 3.1416 rad: the straight angle.
- 1 rad
- · ≈ 57.296°: the angle whose arc equals the radius.
Validity domainThe most widespread mistake of the chapter does not come from the conversion but from the mode: a calculator in DEG mode fed an angle in radians (or the reverse) returns a wrong result with no warning at all.
The most widespread mistake of the chapter does not come from the conversion but from the mode: a calculator in DEG mode fed an angle in radians (or the reverse) returns a wrong result with no warning at all. The classic symptom: sin(30) returning −0.988 instead of 0.5, because the machine read 30 radians. Before any trigonometric calculation, check the displayed mode; when in doubt, sin(180) must return 0 in degrees.
Why mathematics prefers the radianEvery formula of calculus assumes angles in radians: the derivative of sin(x) equals cos(x) only when x is in radians, arc length equals r × θ only in radians, and the small-angle approximation sin(θ) ≈ θ only makes sense in radians.
Every formula of calculus assumes angles in radians: the derivative of sin(x) equals cos(x) only when x is in radians, arc length equals r × θ only in radians, and the small-angle approximation sin(θ) ≈ θ only makes sense in radians. The degree remains the unit of descriptive geometry and instrument readings; the radian, the unit of calculation.