To convert an angle into radians, multiply its degree value by π/180, because 180° is exactly π radians by definition. That single factor handles every degree-to-radian conversion: 30° becomes π/6 ≈ 0.5236 rad, 90° becomes π/2 ≈ 1.5708 rad, and 360° becomes 2π ≈ 6.2832 rad. The reverse direction, radians to degrees, multiplies by 180/π, so one radian equals roughly 57.2958°. Beyond degrees, radians are one of six common angle units — alongside gradians, turns, arcminutes, and arcseconds — and a proper converter should give you every one of those at once so you can cross-check the result. For routine problems, plugging the degree value into an angle converter is faster than reaching for π by hand and removes the risk of a misplaced constant, especially for non-round inputs like 47° or −112.5° where the answer is not a tidy fraction of π. The conversion is exact, not approximate: 180° really does equal π radians to infinite precision, so a result shown to about eight significant figures is more than enough for engineering work, trigonometry, physics problems, and computer-graphics calculations.
Defining the Radian and the Degree
An angle is just the size of a rotation, but the number you read depends on which unit you used to count it. Degrees, radians, gradians, turns, arcminutes, and arcseconds all describe the same rotation — they differ only in how many increments they pack into one full circle. Converting an angle into radians means rewriting that same rotation using the SI unit for plane angle, where one full circle is 2π radians and a half-circle (180°) is exactly π.
The key insight is that the actual rotation never changes when the unit does. A right angle is 90°, π/2 radians, 100 gradians, a quarter turn, 5,400 arcminutes, and 324,000 arcseconds — six different numbers, one rotation. Once that lands, angle conversion reduces to multiplication by a fixed constant.
That is why every textbook and reference starts with the same fact: 180° = π radians. From that single equality, every degree↔radian conversion you'll ever need follows without memorizing a separate factor. A detail worth holding onto: zero converts to zero in every unit, a negative angle represents rotation in the opposite direction, and converting a value to its own unit returns it unchanged — these three properties hold across the entire family of angle measures, not just degrees and radians.
The Core Conversion Factor
Starting from 180° = π radians, divide both sides by 180 to get 1° = π/180 radians. That fraction is the conversion factor. To turn any degree value into radians, multiply by π/180:
radians = degrees × π/180
To go the other way, divide by π/180 (equivalently, multiply by 180/π):
degrees = radians × 180/π
Both formulas are exact. They never round, no matter how big or small the input. For quick mental math, 1 radian ≈ 57.2958° is a useful reference and 1° ≈ 0.017453 radians is its mirror image. Those numbers come from the same π/180 ratio, just inverted, so they always agree with the formulas above.
| Unit | Symbol | Per full circle | Equivalent of 1° |
|---|---|---|---|
| Degree | ° | 360° | 1° |
| Radian | rad | 2π rad | π/180 rad |
| Gradian (gon) | gon | 400 gon | 10/9 gon |
| Turn (revolution) | tr | 1 turn | 1/360 turn |
| Arcminute | ′ | 21,600′ | 60′ |
| Arcsecond | ″ | 1,296,000″ | 3,600″ |
The full-circle column shows what one complete rotation looks like in each unit, and the 1° column is the factor you'd multiply a degree value by to express it in that unit. The radian row is the one this article focuses on, but the others come up often enough that the converter covers all six at the same time.
Converting an Angle Into Radians Using the Angle Converter
For routine work, a calculator-style tool removes the slip-up risk of doing π/180 by hand. The Angle Converter takes any angle, in any of the six units in the table above, and returns it in any other unit — plus the full table of every other unit underneath.
- Type the angle you want to convert into the Value field. Decimals and negative numbers are accepted, so values like 47.5 or −112.5 work as written.
- Choose the source unit under "From" and the target unit under "To." Use the Swap control to flip them if you selected them the wrong way around.
- Read the converted result — it updates as you type — and use the table beneath to see the same angle expressed in every other supported unit at the same time.
Results show to about eight significant figures, so common reference values stay readable: 180° displays as 3.1415927 radians rather than a noisy floating-point string. Very large or very small inputs switch to compact scientific notation so the answer remains legible. Everything runs locally in your browser, which means the conversion is private, works without a server round-trip, and continues to function once the page is loaded even if you go offline. There is no sign-up and nothing is uploaded.
A sensible pattern is to set "From" to the unit your source value is already in, set "To" to whatever the problem requires, and let the all-units table act as a free sanity check — if the converter says 90° is 1.5707963 radians and π/2 ≈ 1.5707963, the math agrees with you.
A Worked Example: 47° in Radians
Not every conversion lands on a clean fraction of π. Take 47°. Substituting into the formula:
radians = 47 × π/180 = 47π/180 ≈ 0.82030475 rad
Plug 47 into the Value field, set "From" to Degrees and "To" to Radians, and the converter returns 0.82030475 (eight significant figures). Because the full-units table sits right under the result, you can also see that 47° is the same as 52.2222… gradians, 0.13055… turns, 2,820 arcminutes, and 169,200 arcseconds — useful when a problem statement mixes units, or when you want to confirm a surveying or astronomy reading without retyping the value into a separate tool.
Try the reverse by swapping "From" and "To" with one click: 0.82030475 radians returns 47° unchanged. Converting a value to its own unit is a quick way to verify the converter is doing what you expect before trusting it on the real problem.
From Radians Back to Degrees
Sometimes the input is already in radians and the output should be degrees — for example when reading off a chart label, working a navigation bearing in an old map, or reporting an angle to a colleague who only thinks in degrees. The same tool handles this direction without needing a second interface.
Set "From" to Radians and "To" to Degrees, type the radian value, and read the equivalent in degrees. Using the inverse factor, 1 radian ≈ 57.2958°, so 2 radians ≈ 114.5916° and π/2 ≈ 1.5707963 radians ≈ 90°. The relationship is symmetric under inversion: 180° = π radians implies 180/π degrees per radian, and every value's conversion is exact to the eight-figure display. Where the result truly is a clean multiple of π — π/6, π/4, π/3, π/2 — the converter prints the unrounded radian, and its degree reading is a tight, rounded multiple of 30°, 45°, 60°, or 90°.
When Math and Physics Insist on Radians
Several standard formulas only work when the angle is in radians. The derivative of sin(x) is cos(x), and the derivative of cos(x) is −sin(x) — but only when x is in radians. The Taylor series sin(x) = x − x³/3! + x⁵/5! − … carries the same restriction; with degrees, every term picks up an extra π factor that clutters the algebra. That is why calculators and programming languages default to radians in their trig functions, and why a degree-mode setting is almost always a button you have to press manually.
Physics and engineering use radians for the same reason. Angular speed ω in rad/s, paired with a radius r, gives a linear speed v = rω directly — no extra 180/π conversion factor and no hidden scale. Computer graphics uses radians for the same reason the math does: rotation matrices, quaternions, and shader-level trig functions all expect their argument in radians, and feeding them degrees silently produces wrong rotations that are very hard to debug.
Surveying is the one place where degrees dominate so heavily that radians almost never appear, and the same is true for navigation bearings, compass headings, and most map labels. Astronomy works at the other end of the scale, where degrees break into arcminutes (1° = 60′) and arcseconds (1′ = 60″, so 1° = 3,600″) — a single arcsecond is roughly the angular size of a coin seen from about three kilometres. For star separation, telescope resolution, GPS coordinate work, and cartography, the all-units view is genuinely useful because a single angle can shift between scales without rounding noise.
Whether your input is a textbook angle, a navigation bearing, a star coordinate, or a gear ratio, the conversion math is the same: multiply by π/180 to go to radians, multiply by 180/π to come back, backed by the fixed equivalences 360° = 2π rad = 400 gon = 1 turn = 21,600′ = 1,296,000″. Once that core relationship is in your head, a tool like the Angle Converter is the fastest way to handle the long tail of non-round inputs — and seeing the full table of all six units at once catches the misreads a single-result calculator can hide.