To convert any angle to degrees, multiply the value by the exact factor that ties that unit to the 360° full circle: radians go through 180/π, gradians through 0.9, turns through 360, arcminutes through 1/60, and arcseconds through 1/3600. These factors are not approximations to memorize — they come straight from the fixed relationships 180° = π radians, 360° = 400 gradians = 1 turn, and 1° = 60 arcminutes = 3600 arcseconds. The arithmetic is identical every time, which is precisely why a dedicated angle converter is the safest tool for the job. Typing the value, choosing the source unit, and reading the result removes the risk of swapping a factor or losing a decimal. For one-off conversions a hand calculator is fine; for the common case where you also want to see the same angle in every other unit at once, the free Angle Converter shows all six side by side, computed locally in your browser with no upload and no signup.
The Six Angle Units the Converter Handles
The converter works between six units that together cover everyday geometry, scientific computing, surveying and astronomy. Degrees (°) are the familiar unit where a full circle is 360; this is the everyday unit for navigation, geometry and screen rotation, and 1° is 1/360 of one complete revolution. Radians (rad) are the SI unit for plane angle and are defined so that an arc equal in length to the radius subtends exactly one radian; because they tie angle directly to arc length, they are the natural choice in calculus, trigonometry, physics and computer graphics. Gradians, also written gon or grad, divide the right angle into 100 units so a full circle is 400 gon — a base-10 scheme that keeps quadrant arithmetic clean and survives in surveying, geodesy and civil engineering, mainly in parts of Europe. Turns, sometimes called revolutions, are simply one complete rotation, equivalent to 360°, 2π radians, or 400 gon, and are useful for describing gears, wheels, phases and any periodic motion in whole cycles. Arcminutes (′) and arcseconds (″) subdivide the degree into the small angles used in astronomy, optics, telescope resolution, GPS positioning and cartography.
Exact Conversion Factors to Degrees
Because the relationships between units are fixed, the factor to multiply by is also fixed. The table below lists the exact factor for converting one unit of each supported type into degrees. Apply them directly and you will get the same number the converter displays.
| Unit | Symbol | Multiply by | Origin |
|---|---|---|---|
| Degree | ° | 1 | Identity |
| Radian | rad | 180/π (≈ 57.29577951) | 180° = π rad |
| Gradian (gon) | gon | 0.9 | 400 gon = 360° |
| Turn | rev | 360 | 1 revolution = 360° |
| Arcminute | ′ | 1/60 (≈ 0.01666667) | 60′ = 1° |
| Arcsecond | ″ | 1/3600 (≈ 0.00027778) | 3600″ = 1° |
All six factors are exact algebraic values; the decimals shown after the slash are simply printed approximations that help when you need to sanity-check a result on paper. To work the other direction — degrees into another unit — divide by the same factor or multiply by its reciprocal. For example, degrees × π/180 gives radians, and degrees × 1/360 gives turns.
Convert an Angle to Degrees in Three Steps
When you have an angle expressed in radians, gradians, turns, arcminutes or arcseconds and need it in degrees, the shortest reliable path is to let the converter do the multiplication. The whole workflow runs in your browser, updates as you type, and accepts decimals and negative numbers.
- Type the angle you want to convert into the Value field — decimals and negative numbers are allowed. Enter 2 if the value is two radians, or 250 if it is 250 gradians.
- Choose the source unit under "From" and the target unit under "To" — use the Swap control to reverse the two if you change your mind.
- Read the converted result instantly, plus a table that shows the same angle in every unit at once. Copy the degree figure you need; if you want to convert the other direction later, Swap and the same value reappears on the new pair.
The converter applies the exact factors from the table above, and results are shown to about eight significant figures, so common conversions read cleanly — for instance, 180° displays as 3.1415927 radians rather than a noisy floating-point string, and 90° displays as π/2 ≈ 1.5707963 radians. Results are shown to about eight significant figures, and very large or very small values switch to compact scientific notation so the figure stays readable rather than overflowing the cell.
Where Each Unit Tends to Show Up
Different disciplines settle on different angle units for historical and practical reasons. Picking the right source unit before you convert avoids accidentally working in the wrong scale.
| Unit | Most common in | Reason it is used there |
|---|---|---|
| Degrees | Navigation, geometry, screen rotation, everyday measurement | Base-60 subdivisions, familiar 360° circle, easy to read on a protractor |
| Radians | Calculus, trigonometry, physics, computer graphics, programming | SI unit, links angle directly to arc length, derivatives of sine and cosine are clean only in radians |
| Gradians | Surveying, geodesy, civil engineering (mainly in parts of Europe) | Right angle is exactly 100 gon, so quadrant arithmetic is base-10 |
| Turns | Gears, wheels, rotating machinery, periodic motion, animation phases | Whole cycles read as whole numbers instead of multiples of 360 |
| Arcminutes | Astronomy, optics, GPS, cartography, surveying | One degree splits into 60 for the sub-degree precision those fields need |
| Arcseconds | Astronomy, telescope resolution, satellite tracking, fine optics | One degree splits into 3600 for arc-level precision over sky and distance |
The unit you start in is almost always the unit the problem statement, textbook or hardware gave you. Converting to degrees lets you compare across these domains on a familiar scale, then convert back into the discipline's preferred unit for the actual calculation. If your input happens to be a GPS reading rather than a bare angle, our guide on converting GPS coordinates to degrees, minutes and seconds walks through the same factors applied to a coordinate pair.
Two Radians in Degrees: A Worked Example
To make the formula concrete, here is a single worked conversion from radians to degrees. Suppose a calculation gives an angle of 2 radians and you need the equivalent in degrees for a report or a plot label.
The formula is degrees = radians × 180/π, which comes directly from the fixed identity π radians = 180°.
Substituting the value: 2 rad × 180/π = 360/π.
Evaluating the division using π ≈ 3.14159265358979: 360 / 3.14159265358979 ≈ 114.5915590.
So 2 radians equals approximately 114.5916°. Reading off the same value in the other direction — what 114.5916° looks like in radians — gives back 2, confirming the conversion is exact up to the eight-figure display limit of the tool. For the inverse case, 1 radian × 180/π ≈ 57.2958°, a number worth remembering because it shows up whenever you translate a textbook problem from radians into degrees.
Negative Angles, Zero, and Very Large Values
Negative angles are perfectly valid and represent rotation in the opposite direction; the same factor applies, the sign carries through, and the converter accepts them in the Value field just like positive numbers. Zero converts to zero in every unit, so an angle that has been "reduced" to zero from a long expression will read identically across the table. Converting a value to its own unit — degrees to degrees, radians to radians — returns the value unchanged, which is a useful sanity check that you picked the right source and target.
Results are shown to about eight significant figures, which is enough for engineering, surveying, navigation and most physics problems. Very large angles (think gear ratios in the millions of turns) and very small angles (think micro-arcsecond telescope pointing) switch automatically to compact scientific notation so the figure stays readable rather than overflowing the cell. Because every conversion is computed locally in the browser with no upload and no server round-trip, the tool remains fast and private once the page has loaded.
Reading the Full Conversion Table at a Glance
A small but very useful feature of the converter is the side table that lists the same angle in every supported unit at the same time. When you are working on a problem that crosses disciplines — say, a camera angle given in turns that needs to land as gradians for a surveying spreadsheet and as arcminutes for a telescope pointing file — you do not have to run six separate conversions. Type the value once, set the source unit, and the table fills in degrees, radians, gradians, turns, arcminutes and arcseconds for that single value. From there you copy whichever figures your downstream step needs.
This is also how the converter doubles as a cross-check: if your hand calculation for one of the units disagrees with the table, the error is almost certainly in the factor you used, not the typed value. For the everyday case of converting a single value from one unit into degrees, the three-step workflow above is all you need; the full table is the safety net for the harder jobs and the moment when a conversion crosses from one field into another.
If you're weighing options, Accurate Angle Converter: 6 Units, Exact Factors covers this in detail.