Degrees vs Radians
Getting this toggle wrong is the most common source of wrong trigonometry answers. Here is what each unit means and when to use which.
Last updated: August 2026 · Written and verified by Akash Pandey
Two ways to measure the same angle
A degree is one 360th of a full turn. The choice of 360 is historical - it descends from Babylonian base-60 arithmetic and the convenience of a number with a great many divisors. There is nothing mathematically special about it.
A radian is defined by the circle itself: it is the angle subtended at the centre by an arc equal in length to the radius. Since a full circumference is 2πr, a full turn is exactly 2π radians. That makes the radian a ratio of two lengths, and therefore dimensionless.
radians = degrees × π / 180 degrees = radians × 180 / π
- π radians = 180°, which is the only conversion fact worth memorising.
- One radian ≈ 57.29578°.
- One degree ≈ 0.0174533 radians.
Conversion table
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.523599 |
| 45° | π/4 | 0.785398 |
| 60° | π/3 | 1.047198 |
| 90° | π/2 | 1.570796 |
| 120° | 2π/3 | 2.094395 |
| 135° | 3π/4 | 2.356194 |
| 180° | π | 3.141593 |
| 270° | 3π/2 | 4.712389 |
| 360° | 2π | 6.283185 |
Why calculus insists on radians
The derivative of sin x is cos x - but only when x is in radians. In degrees it picks up an awkward constant factor of π/180, and every subsequent derivative accumulates another. Every Taylor series for a trigonometric function assumes radians for the same reason.
The underlying fact is the small-angle limit: as x approaches 0, sin(x)/x approaches exactly 1 in radians. In degrees that limit is π/180 ≈ 0.01745. Radians are the unit in which the derivative comes out clean, which is why they are the default in every programming language and mathematics library, including the one underneath this calculator.
Which should you use?
- Degrees - geometry, surveying, navigation, construction, most physics problems stated in degrees, and school trigonometry.
- Radians - calculus, any use of a Taylor series, angular velocity and simple harmonic motion, complex numbers and Euler's formula, and any programming language's built-in trig functions.
A floating-point footnote
In degree mode, tan(90) should be undefined - the tangent has a vertical asymptote there. What the calculator actually returns is an enormous finite number, around 1.633 × 10¹⁶.
The reason is that 90° is converted to radians by multiplying by π/180, and the result is the nearest representable double to π/2 rather than π/2 itself. The tangent of a value a hair away from the asymptote is very large but perfectly finite. No calculator working in binary floating point can avoid this; the honest reading of such a result is "undefined", and a value of that magnitude appearing out of a tangent is the signal. The same applies to tan(270).
Frequently Asked Questions
How do I convert degrees to radians?
Multiply by π/180. So 60° becomes 60 × π/180 = π/3 ≈ 1.047198 radians.
How do I convert radians to degrees?
Multiply by 180/π. So π/4 radians becomes 0.785398 × 180/π = 45°.
How many radians are in a circle?
Exactly 2π, which is about 6.283185. A half turn is π radians, or 180°.
Why does my calculator give the wrong answer for sin(30)?
Almost certainly because it is in radian mode. sin(30) is 0.5 in degrees but about −0.988 in radians. Switch the DEG/RAD chip and try again.
Which mode should I leave the calculator in?
Degrees for school geometry and most everyday problems; radians for calculus and anything involving a series expansion. This calculator remembers your choice between visits.
Is a radian a unit like a metre?
Not quite. It is a ratio of an arc length to a radius, so the units cancel and it is dimensionless. That is precisely why it drops cleanly out of derivatives.