To convert an angle to radians, multiply the number of degrees by π/180, or equivalently divide by 57.2958; this single factor comes from the fixed relationship 180° = π radians exactly, so degrees × π/180 gives radians, radians × 180/π gives degrees, and 1 radian measures about 57.2958°. The conversion is exact in theory and shown to about eight significant figures in practice, so 180° displays as 3.1415927 rather than a noisy floating-point string. Radians sit at the center of trigonometry, calculus and physics because they tie angle to arc length on a circle: an arc equal in length to the radius always subtends exactly one radian, which is why derivatives of sine and cosine simplify cleanly only when the input is in radians. Most everyday problems still arrive in degrees, which is why a reliable browser tool is the simplest way to swap units without re-keying π by hand. The Angle Converter handles this for degrees, radians, gradians, turns, arcminutes and arcseconds in one place, using the exact factors built into the tool.
The Degrees-to-Radians Formula
The whole conversion rests on one identity: 180° = π radians. From that single equality, every other relationship follows by simple arithmetic. To go from degrees to radians, you multiply degrees by π/180. To go from radians to degrees, you multiply radians by 180/π. Because π is irrational, the result is almost never a clean whole number, but the factor itself is exact.
Practically, three equivalent statements cover most of what you will ever do:
- Degrees to radians: rad = deg × π / 180
- Radians to degrees: deg = rad × 180 / π
- 1 radian = 180/π degrees ≈ 57.2958°
If you remember only one number, remember 57.2958. To convert any radian value to degrees, multiply by 57.2958; to go the other way, divide by 57.2958. This shortcut matches the formula exactly because 180/π rounds to 57.29577951…, and the tool rounds to eight significant figures, so the difference between using π and using 57.2958 is invisible for ordinary problem-solving.
A subtle but useful point: 0 in any unit converts to 0 in every other unit. Converting a value to its own unit returns it unchanged. Negative angles are valid in every unit and represent rotation in the opposite direction. These rules are built into the converter so you never get a surprise sign flip or a non-zero result from a zero input.
How to Convert an Angle to Radians with the Angle Converter
For day-to-day work, especially when π is hard to key in or the problem mixes units, the Angle Converter handles the arithmetic for you. Here is the exact sequence the tool runs through:
- Open the Angle Converter in your browser. No sign-up, no upload — the page computes locally.
- Type the angle you want to convert into the Value field. Whole numbers, decimals and negative numbers all work, and the result updates as you type.
- Pick the source unit under "From" — for example Degrees if you are starting from a textbook angle in °.
- Pick the target unit under "To" — for example Radians. The two dropdowns are independent, so any unit can convert to any other unit, including the same one.
- If you need the reverse direction later, use the Swap control to flip From and To in a single click without retyping the value.
- Read the converted result shown by the tool. For a degrees-to-radians conversion, the result is shown to about eight significant figures, so 180° reads as 3.1415927 rad rather than a long noisy floating-point string.
- Scan the table at the bottom of the tool for the same angle expressed in every supported unit at once: degrees, radians, gradians, turns, arcminutes and arcseconds. This is handy when a problem needs the angle in more than one unit — say degrees for a plot label and radians for the formula underneath it.
Everything runs in your browser once the page has loaded, so the tool keeps working offline and your input never leaves the device.
Worked Example: Converting 45° to Radians
Take a single concrete case and walk it through by hand, so the formula is no longer abstract.
The formula is: radians = degrees × π / 180.
Substitute the number: radians = 45 × π / 180.
Simplify the fraction: 45 / 180 = 1/4, so the expression becomes π / 4.
Evaluate: π / 4 ≈ 3.14159265 / 4 ≈ 0.78539816.
Therefore 45° = π/4 radians, which the tool rounds and displays as 0.78539816 rad. If you instead start with π/4 and want degrees, multiply by 180/π: π/4 × 180/π = 45, recovering the original value with no rounding loss in either direction.
Angle Units and Their Radians Equivalent
The tool converts between six units, all anchored to the same circle. The table below lists each unit and its exact factor relative to one radian, so you can see at a glance how the units compare.
| Unit | Symbol | Equivalent in radians (exact factor) |
|---|---|---|
| Degree | ° | π / 180 ≈ 0.01745329 rad |
| Radian | rad | 1 rad |
| Grad / gon | gon | π / 200 ≈ 0.01570796 rad |
| Turn / revolution | rev | 2π ≈ 6.28318531 rad |
| Arcminute | ′ | π / 10800 ≈ 0.000290888 rad |
| Arcsecond | ″ | π / 648000 ≈ 0.000004848 rad |
These factors are fixed by definition: a degree is 1/360 of a full circle, a radian is defined by arc length equal to the radius, a gradian puts 400 units in a full circle, a turn is one whole revolution, an arcminute is 1/60 of a degree and an arcsecond is 1/60 of an arcminute (so 3600 arcseconds per degree). Because every other unit in the table is defined against the degree, and the degree is fixed against the radian by 180° = π, the chain of conversions is exact.
Why Radians, Not Degrees, in Math and Physics
Radians are the SI unit for plane angle, but the practical reason they dominate calculus and physics is that the radian connects angle to arc length by construction. If you walk around a circle of radius r, the arc length you cover equals r × θ when θ is in radians — no extra constant required. The same link makes the small-angle approximation sin(θ) ≈ θ and tan(θ) ≈ θ work to high accuracy for θ near zero, but only when θ is in radians.
Trigonometric derivatives clean up only in radians as well. The derivative of sin(x) with respect to x is cos(x) if and only if x is in radians. In degrees, an extra factor of π/180 is needed and the derivative is not even a clean trig function of the original input. Power series for sine and cosine, Fourier transforms, orbital mechanics, rotational kinematics, complex exponentials and most physics formulas are all written assuming radians.
Degrees survive because they divide the circle into 360 familiar pieces, a number tied to ancient calendars and easy to subdivide. Gradians (also called gons) divide the circle into 400 units so a right angle is exactly 100 gon, which keeps quadrant arithmetic clean; they survive mainly in surveying, geodesy and civil engineering in parts of Europe. Turns are handy for whole revolutions — gears, wheels, phases, periodic motion — and arcminutes plus arcseconds subdivide the degree for fine work in astronomy, optics, GPS positioning and cartography.
Common Conversions the Tool Covers Instantly
A few conversions come up over and over, and the Angle Converter handles each without retyping π:
- Degrees to radians for a physics problem: 90° = π/2 ≈ 1.5708 rad, 30° = π/6 ≈ 0.5236 rad.
- Radians to degrees for a plot label or report: π/3 ≈ 1.0472 rad becomes 60°.
- Degrees to arcseconds for an astronomy calculation: 1° = 3600″ exactly.
- Turns to radians for rotational mechanics: 1 turn = 2π ≈ 6.2832 rad.
- Gradians to degrees for a surveying input: 100 gon = 90° exactly.
For each of these, the converter displays the same angle in all six units at once, so you can copy the value you need without running a second conversion. Very large or very small values switch to compact scientific notation, while common angles read cleanly to eight significant figures.
When to Convert by Hand and When to Use the Tool
By-hand conversion is fine for clean multiples of 30°, 45°, 60° and 90°, because the π factor cancels and you land on a tidy fraction of π. For anything else — a survey bearing of 37.42°, an angle of 2.7 rad, a telescope offset of 0.013° — the arithmetic gets messy and a small rounding error compounds if you copy the result into another formula. The Angle Converter keeps eight significant figures and uses the exact π-based factors internally, so the numbers line up with the formulas you paste them into.
For a class problem or a quick check, the tool is also faster than reaching for a separate calculator, and because the result updates as you type, you can sweep through a range of angles without re-entering each one. Because everything is computed locally with no upload and no server round-trip, the tool is private, responsive, and keeps working offline once the page has loaded. The same exact factors apply whether you start in degrees, gradians, turns or any other supported unit, so the numbers you copy into downstream formulas line up with the textbook values.