Degrees vs radians: the 5-second check
By MathSolver Editorial Team · Published October 7, 2026 · Updated October 23, 2026
Published October 2026.
Degrees split a circle into 360 parts (an ancient Babylonian habit). Radians measure angle by arc length: a full circle is 2π radians. Same angle, two languages.
When is each used?
Geometry and everyday problems use degrees. Calculus, physics formulas, and programming libraries default to radians, derivatives like d/dx sin x = cos x are only true in radians.
The 5-second check
Before trusting any trig answer, compute sin 30° mentally: it must be 0.5. If your tool says −0.988, you are in radian mode. Our trig calculator lets you pick the unit explicitly, so the mode can never surprise you.
The two conversion formulas
One fact connects the two units: 180 degrees equals π radians. Every conversion is just a scaling of that fact, so you only need two formulas.
Degrees to radians: radians = degrees × (π / 180)
Radians to degrees: degrees = radians × (180 / π)
Worked example 1: Convert 45 degrees to radians. Multiply by π / 180: 45 × π / 180 = π / 4, which is about 0.785 radians.
Worked example 2: Convert 120 degrees to radians. 120 × π / 180 = 2π / 3, which is about 2.094 radians.
Worked example 3: Convert 2 radians to degrees. Since 180 / π is about 57.2958, multiply: 2 × 57.2958 = about 114.59 degrees.
Keep one sanity check in mind. Radian measures of everyday angles are small numbers, somewhere between 0 and about 6.283 for anything up to a full turn. If your radian answer comes out in the hundreds, you converted in the wrong direction.
Common angles at a glance
| Degrees | Radians (exact) | Radians (approx) |
|---|---|---|
| 0 | 0 | 0 |
| 30 | π/6 | 0.524 |
| 45 | π/4 | 0.785 |
| 60 | π/3 | 1.047 |
| 90 | π/2 | 1.571 |
| 180 | π | 3.142 |
| 270 | 3π/2 | 4.712 |
| 360 | 2π | 6.283 |
Look at the pattern. Every radian value in the table is a simple multiple or fraction of π, built from halves, thirds, quarters and sixths. If you memorize the first quadrant only (30, 45, 60 and 90 degrees), you can rebuild almost every angle a textbook will throw at you by adding or subtracting those.
Where each unit earns its keep
In daily life, degrees win because they are easy to say, sketch and measure. A protractor is marked in degrees. A compass bearing is in degrees, with north at 0, east at 90, south at 180 and west at 270. Weather reports, map directions and school geometry all speak degrees, mostly in friendly whole numbers.
In science, the radian does real work that degrees cannot. The arc length formula, s = r × θ, only gives the right answer when θ is in radians. Try it: a circle has radius 5 cm and the angle is π/3 radians, so s = 5 × π/3 = 5π/3, about 5.24 cm of arc. Now plug the same angle in as 60, its degree measure, without converting. The formula returns 300, which is nonsense for a circle only 5 cm in radius. Physics, engineering and calculus stick to radians because their formulas were built on the arc length definition, and degrees quietly break them.
Three mistakes that cost marks
1. Converting the wrong way. If you turn 60 degrees into radians and get about 3438, you multiplied by 180 / π instead of π / 180. Ask which answer should be smaller. Radian numbers for ordinary angles are small, so the degree number almost always shrinks in the conversion.
2. Feeding degrees into radian formulas. Arc length, sector area and every calculus formula assume radians. Convert the angle first, then calculate. The 5 cm circle above shows how badly it goes when you skip that step.
3. Dropping the π. π/3 is about 1.047, while plain 1/3 is 0.333. They are very different angles. If a problem expects an exact answer, keep the π in your result instead of rounding it away early.
Frequently asked questions
How big is one radian?
About 57.3 degrees, so it is a much bigger step than one degree. A radian is the angle you get when the arc length along the circle equals the radius. A full circle has circumference 2π times the radius, which is why it holds exactly 2π radians.
How can I convert in my head?
Multiply degrees by about 0.0175 to get radians: 90 × 0.0175 is about 1.58, close to the exact π/2 value of 1.571. Going the other way, multiply radians by about 57.3: 1.5 × 57.3 is about 86, and 1.5 radians really is about 85.94 degrees.
Is the radian a real unit?
Sort of. A radian is a ratio of two lengths, arc divided by radius, so the lengths cancel and the radian carries no dimension of its own. A degree is a pure counting unit instead: someone simply decided long ago that a circle should have 360 of them.
Which unit should my calculator use?
Match the unit in the question. If the angle carries a degree symbol, work in degree mode. If the angle is written with π, or appears as a bare number inside a calculus or physics problem, work in radian mode. Converting inputs to one unit before you start prevents nearly every mix-up.