Degrees to Radians Converter

π radians is 180 degrees — convert angles for trigonometry, calculus and code, with the exact multiples of π.

Conversion table

Angle	In radians
1	0.017453 rad
2	0.034907 rad
3	0.05236 rad
4	0.069813 rad
5	0.087266 rad
10	0.17453 rad
15	0.2618 rad
20	0.34907 rad
25	0.43633 rad
30	0.5236 rad
40	0.69813 rad
50	0.87266 rad
60	1.0472 rad
70	1.2217 rad
75	1.309 rad

🔢 The multiples of π

30° = π/60.5236 rad
45° = π/40.7854 rad
90° = π/21.5708 rad
1 rad57.2958°

Working with angles without slips

Multiply degrees by π/180 to get radians, and by 180/π to come back. Written as a single number: × 0.0174533 and × 57.29578.
Leave the answer as a multiple of π whenever you can. 3π/4 is exact; 2.356 is a rounded copy of it that will drift through a long calculation.
Check your calculator's mode before every trig question. sin(30) is 0.5 in DEG and −0.988 in RAD, and both look like plausible answers.
Every programming language's sin, cos and tan take radians. In JavaScript and Python the conversion is angle * Math.PI / 180.

What a radian actually measures

Take a circle, walk along its edge until the arc you have covered is exactly as long as the radius, and look back at the angle you have swept. That angle is one radian. Because the full circumference is 2πr, a whole turn contains exactly 2π radians — about 6.2832 — and half a turn contains π. This is why 180° = π rad, and why one radian is 180/π ≈ 57.2958 degrees. Unlike the degree, the radian is not a chosen number: it comes out of the geometry of the circle itself, which is why it is the SI unit for plane angle.

Why calculus insists on radians

In radians, arc length is simply s = rθ and sector area is ½r²θ, with no conversion factor. More importantly, the derivative of sin x is cos x only when x is in radians; in degrees the derivative picks up a factor of π/180 that then infects every formula built on it. The same choice makes the small-angle approximation work: for a small θ in radians, sin θ ≈ θ ≈ tan θ, accurate to better than 0.5% below 10°. Physics, optics and signal processing all lean on that shortcut, which is why the degree stays on the protractor and the radian goes into the equations.

The angles worth memorising

30° = π/6 ≈ 0.5236 rad. Sine is exactly 1/2.
45° = π/4 ≈ 0.7854 rad. Sine and cosine are both √2/2.
60° = π/3 ≈ 1.0472 rad. Cosine is exactly 1/2.
90° = π/2 ≈ 1.5708 rad. Sine is 1 and tangent is undefined.
180° = π ≈ 3.1416 rad, and 270° = 3π/2 ≈ 4.7124 rad.
360° = 2π ≈ 6.2832 rad, one full turn.

Where angle conversions go wrong

Calculator in the wrong mode. This is the single most common lost mark in trigonometry, and it never produces an error message — only a wrong number that looks fine.
Rounding π too early. Using 3.14 instead of π costs about 0.05% straight away, and that grows through repeated multiplication.
Mixing gradians in. Some calculators offer a third GRAD mode where a right angle is 100, not 90 or π/2. Landing there by accident is easy.
Feeding degrees to a library function. Math.sin(90) returns 0.894, not 1, because the language read 90 radians.
Forgetting that angles wrap. 390° and 30° are the same direction, as are 2π+x and x.
Confusing arcminutes with minutes of time. 1′ is 1/60 of a degree; it has nothing to do with clocks, though both come from the same Babylonian base-60.

Worked examples

Both directions, exact and decimal

Convert 135° to radians

135 × π/180 = 135π/180
Simplify the fraction: 3π/4
As a decimal: 3 × 3.14159 ÷ 4 = 2.3562 rad

135° = 3π/4 ≈ 2.3562 rad

Convert 2 radians to degrees

2 × 180/π = 360/3.14159
= 114.59°
Sanity check: 2 rad is a bit under π, so a bit under 180°

2 rad = 114.59°

Frequently Asked Questions

How do I convert degrees to radians?

Multiply by π/180. So 90° becomes 90π/180 = π/2 ≈ 1.5708 rad, and 60° becomes π/3 ≈ 1.0472. Going back, multiply radians by 180/π, which is about 57.2958 degrees per radian.

Why is π radians equal to 180 degrees?

Because a radian is defined by arc length. The arc of a half circle is πr, and one radian is the angle whose arc equals one radius, so a half circle holds exactly π radians. Since a half circle is also 180 degrees, the two descriptions of the same angle give π rad = 180°.

What is 1 radian in degrees?

57.29578 degrees, or 180 divided by π. It is a slightly awkward number precisely because the radian is defined by the circle and the degree is defined by a historical choice of 360 parts; nothing forces the two to line up neatly.

Why do mathematicians use radians instead of degrees?

Because the formulas come out clean. Arc length is rθ and sector area is ½r²θ only in radians, the derivative of sin x is cos x only in radians, and sin θ ≈ θ for small angles only in radians. Using degrees leaves a factor of π/180 attached to every derivative and series.

What is a gradian and why does my calculator have one?

A gradian, also called a gon, divides a right angle into 100 parts instead of 90, so a full turn is 400 gon. It was a metric-era attempt to decimalise angles and still survives in surveying, especially in France. If your trig answers are subtly wrong, check that the calculator has not landed in GRAD.

Does the conversion work for negative angles?

Yes, and negative angles are ordinary: they simply turn clockwise instead of counterclockwise. −90° is −π/2 rad, and it names the same direction as +270°. This converter accepts them without complaint.