One degree equals π/180 radians, so any angle in degrees becomes radians by multiplying by π/180. To turn degrees into radians, take the number of degrees, multiply it by π, then divide by 180 — equivalently, multiply by 0.0174532925199433, which is π divided by 180. Going the other way, multiply radians by 180/π (about 57.29577951) to get degrees. Those two factors are the only formulas you need for the most common angle conversion, and they hold exactly because 180° and π radians describe the same half-turn of a circle. Whether you are plotting a sine curve, labeling a physics answer in SI units, or checking a surveying angle, the degrees-to-radians conversion comes down to that single factor. Once that single factor feels familiar, every other unit pair covered here follows from it, and a calculator becomes a confirmation step rather than a necessity.
If you would rather skip the arithmetic entirely, the Angle Converter handles degrees, radians, gradians, turns, arcminutes and arcseconds in one place. Type a value, pick the source and target units, and the result updates as you type. The same angle also appears in a full reference table so you can read off every equivalent value at once.
The Angle Units You Will Meet
Most everyday angles are written in degrees, which divide a full circle into 360 equal pieces. Each piece is 1/360 of a turn, and degrees are the natural choice for navigation bearings, geometry problems, screen rotation settings and latitude/longitude on consumer maps. The radian is the SI unit for plane angle, defined by geometry rather than by an arbitrary 360: an angle of one radian sweeps an arc whose length equals the radius of the circle. That makes radians the natural language of trigonometry, calculus, physics and computer graphics, because the angle value is directly tied to arc length on a unit circle.
Three more units round out the practical set. A gradian (also called a gon or grad) splits the right angle into 100 pieces, so a full circle is 400 gradians; the base-10 scheme keeps quadrant arithmetic clean and survives in surveying, geodesy and some European civil engineering. A turn — also called a revolution or a cycle — is simply one full rotation, equal to 360°, 2π radians or 400 gradians, and it is the convenient unit for gears, wheels, periodic motion and clock phases. Finally, arcminutes and arcseconds subdivide the degree for fine work: one degree equals 60 arcminutes, and one arcminute equals 60 arcseconds, so an arcsecond is 1/3600 of a degree. Those tiny units dominate astronomy, where a planet's apparent diameter and stellar parallax are quoted in arcseconds, optics, where telescope resolution is limited by diffraction and expressed in arcseconds, GPS positioning, where civilian receivers resolve a few metres of latitude and longitude, and cartography, where map scale and projection errors are tracked in fractions of a degree. Sub-degree precision is also useful in surveying for property boundaries, in target sports for rifle and bow sight adjustment, and in machining for angular tolerance, where a tenth of a degree already costs a measurable amount of material.
The Exact Formulas Behind the Conversion
The core relationships are fixed by geometry and never change:
| Reference | Value |
|---|---|
| Degrees per full circle | 360° |
| Radians per full circle | 2π |
| Gradians per full circle | 400 gon |
| Turns per full circle | 1 turn |
| Arcminutes per degree | 60 arcmin |
| Arcseconds per degree | 3600 arcsec |
| Arcseconds per arcminute | 60 arcsec |
From those anchors, every other conversion follows. The degrees-to-radians factor is π/180, so the formula is radians = degrees × π / 180. Going back, degrees = radians × 180 / π. For gradians, since 400 gon = 360°, you get gon = degrees × 10/9 and degrees = gon × 0.9. For turns, radians = turns × 2π and degrees = turns × 360. Arcminutes and arcseconds scale by their simple 60-based prefixes, the same way minutes and seconds divide an hour.
It is worth memorising the two reference angles that show up everywhere: 180° = π radians, and 90° = π/2 radians. From those, 60° = π/3, 45° = π/4 and 30° = π/6 follow by division. If your answer ever looks like a tidy multiple of π, you almost certainly have the radian form right.
How to Convert Angle Measures to Radians
- Open the Angle Converter in your browser. Nothing is installed and nothing is uploaded — the page runs entirely on your device.
- Type the angle you want to convert into the Value field. Whole numbers, decimals and negative numbers are all accepted, so -45, 0.75 and 720 work the same way.
- Choose the source unit under the "From" dropdown — pick Degrees if you are converting from degrees, Radians if you are starting from radians, or any of the other supported units.
- Choose the target unit under the "To" dropdown. Use the Swap control to reverse the two if you change your mind.
- Read the converted value directly under the result panel. Results are shown to about eight significant figures, so a conversion such as 180° displays as 3.1415927 radians rather than a noisy floating-point string. Very large or very small values switch to compact scientific notation automatically.
- Scroll to the reference table below the result. It lists the same input angle in every supported unit at once, which is useful when you need a value in degrees, radians and gradians side by side for a homework problem or a CAD drawing.
The converter is symmetric: typing the same unit in both From and To returns your original value, and entering 0 in any unit gives 0 in every other unit. Negative angles — used to represent clockwise rotation in trigonometry or below-horizon declinations in astronomy — are preserved with the right sign, so -45° reads as -π/4 radians.
A Worked Example: Converting 75 Degrees to Radians
Suppose a geometry problem gives an angle of 75° and you need it in radians for a trig identity. Apply the formula directly:
radians = 75 × π / 180
Cancel the common factor: 75/180 = 5/12, so the exact answer is 5π/12 radians. As a decimal, that is about 1.3089969 radians. The Angle Converter produces the same result from 75° to radians, and its reference table also shows 75° as 1.3089969 radians, 83.3333 gradians, 0.20833 turns, 4500 arcminutes and 270000 arcseconds. Because the tool uses the exact factors above, the gradians and turns columns line up with their own definitions rather than being rounded from the radian value.
You only get one worked numeric example here, so use the tool directly for any other value — typing a different number into the Value field is all it takes.
Why Math and Physics Prefer Radians
Most calculators and programming languages assume their trig functions take radians, not degrees. That is not an accident: the derivative of sin(x) is cos(x) only when x is in radians, the small-angle approximation sin(x) ≈ x works in radians, and arc length on a circle of radius r equals r·θ only when θ is in radians. The Taylor series sin(x) = x - x³/3! + x⁵/5! - ... only sums cleanly with x in radians, and Euler's formula e^(iθ) = cos(θ) + i·sin(θ) is written with θ in radians for the same reason. Radians therefore remove every awkward conversion factor from calculus and physics, which is why equations such as F = mω²r or v = ωr are written with ω in rad/s rather than in degrees per second. Computer graphics engines follow the same convention: every rotation matrix and quaternion assumes radians internally, and degree inputs are converted to radians before the math begins. Signal processing and Fourier analysis sit on the same convention, since rotating a complex number by θ radians uses the formula e^(iθ) = cos(θ) + i·sin(θ) directly. Even simple things, such as typing sin(30) into a calculator and getting a tiny answer, only become intuitive once you remember the calculator is using radians — sin(30) ≈ -0.988, because 30 radians is many full turns, not 30 degrees.
Degrees, by contrast, are a human-friendly division of the circle into 360 parts that goes back to Babylonian astronomy. They are perfect for navigation, map bearings and screen rotation, and most people read them faster than radians. The art is knowing when each unit earns its keep: keep degrees for everyday angles and for communicating results, switch to radians for anything that flows into a derivative, a power series, or a physics equation.
Once you have the basic factors in mind, switching between units is a single multiplication or a single click in the Angle Converter. The exact ratios above guarantee that whatever tool, textbook, or formula you feed the answer into, the numbers will line up.