Angle Conversion Calculator
Convert between radians, degrees and survey angles in one step
Angle conversion FAQ
Why do most formulas use radians instead of degrees?
In calculus and physics, radians make the math cleaner. When angles are measured in radians, the slope of sin(x) at x = 0 is exactly 1, cos(x) has a simple Taylor series, and relationships like arc length s = r·θ use the angle directly with no extra factors. Degrees, gradians and mils are easier to read for humans, but radians are what the underlying math “prefers”. :contentReference[oaicite:0]{index=0}
What is the exact relationship between degrees and radians?
A full circle is 360° and also 2π radians. From that, 1 rad = 360° ÷ (2π) ≈ 57.29578° and 1° = 2π ÷ 360 ≈ 0.017453 rad. The calculator uses these precise relationships internally and then rounds the displayed result. :contentReference[oaicite:1]{index=1}
Where are arcminutes and arcseconds used in real life?
Arcminutes (′) and arcseconds (″) show up in astronomy, surveying, navigation and mapping. A single degree of angle is 60 arcminutes or 3,600 arcseconds, which gives fine control over pointing telescopes, describing GPS positions or laying out precise bearings on a map grid.
What is the difference between degrees, gradians and mils?
All three divide up a full revolution differently. Degrees split a circle into 360 equal parts, gradians (gons) use 400 “right-angle friendly” parts, and many artillery-style mils use 6,400 parts per turn. Conversions like 1 grad = 0.9° and 1 mil ≈ 0.05625° come from those definitions, and the calculator simply applies the corresponding factors. :contentReference[oaicite:2]{index=2}
Is a revolution the same thing as a circle or turn?
For this calculator, yes. A revolution, circle and turn all mean one full rotation: 360°, 2π rad, 400 gons or 6,400 mils. Internally the page treats “rev” and “cir” as the same unit and converts through radians.
Can this handle negative angles or angles over 360°?
Absolutely. Angles on this page are just numbers passed through conversion formulas. A value like -45° is simply -π/4 rad, and 810° is 2.25 revolutions. If you need to wrap results into the 0–360° or 0–2π range, you can do that after converting.
One calculator for all your angle units
This page is a focused angle conversion calculator. Instead of juggling separate tools for radians, degrees, arcminutes, arcseconds, gradians, revolutions and mils, you can convert between all of them in one place. It is aimed at quick checks for trigonometry homework, engineering sketches, surveying tasks, ballistics and navigation problems.
1. Pick your starting and target units
The dropdowns include the angle units used most often in maths and field work:
- Radians (rad) and milliradians (mrad) for calculus, physics and some range calculations.
- Degrees (°), arcminutes (′) and arcseconds (″) for astronomy, maps and navigation.
- Gradians (gons) for survey and civil engineering where 100 gons form a right angle.
- Revolutions (rev) and circles (cir) for whole turns of a shaft or wheel.
- Mils (6,400 per revolution in this tool) for artillery and some optical sight reticles.
Choose your “Convert from” unit, then your “Convert to” unit. You can flip between any pair: degrees to radians, arcseconds to mils, revolutions to gradians and more.
2. Base-unit method: everything through radians
Under the hood, every value is routed through radians, the SI unit for plane angle. Radians are defined so that a full circle is 2π rad and a right angle is π/2 rad. The calculator uses formulas like: :contentReference[oaicite:3]{index=3}
θrad = θdeg × (π / 180)
θrad = θgrad × (π / 200)
θrad = θrev × 2π
θrad = θmil × (2π / 6400)
Once the angle is in radians, converting to another unit is the reverse:
θdeg = θrad × (180 / π)
θgrad = θrad × (200 / π)
θrev = θrad / (2π)
θmil = θrad × (6400 / 2π)
3. Angle conversion factors at a glance
If you prefer to eyeball numbers or do a quick estimate without hitting Convert, this table lists common angle conversions that the calculator uses internally.
| Starting Unit | Resulting Unit | Conversion Factor |
|---|---|---|
| Radians, degrees and turns | ||
| radians (rad) | degrees (°) | 57.29578 |
| degrees (°) | radians (rad) | 0.017453 |
| revolutions (rev) | radians (rad) | 2π ≈ 6.283185 |
| radians (rad) | revolutions (rev) | 1 / (2π) ≈ 0.159155 |
| Degrees, minutes, seconds | ||
| degrees (°) | arcminutes (′) | 60 |
| degrees (°) | arcseconds (″) | 3,600 |
| arcminutes (′) | radians (rad) | 0.000290888 |
| arcseconds (″) | radians (rad) | 0.0000048481 |
| Gradians and mils | ||
| gradians (gon) | degrees (°) | 0.9 |
| gradians (gon) | radians (rad) | 0.015708 |
| mils | degrees (°) | 0.05625 |
| mils | radians (rad) | 0.000981748 |
4. Reading and using the result
The result box always keeps things simple. If you type 90° and convert to radians, you will see something like “90 ° = 1.570796 rad”. Low-precision tasks such as checking a sketch, reading off a protractor, or converting a mil reading to degrees only need a couple of decimal places. When you are feeding numbers into sensitive simulations, finite-element models or orbital calculations, keep an extra digit or two for your own work and treat this calculator as a quick verification step.
Because everything flows through radians, the tool stays consistent with the way angle is defined in the modern International System of Units (SI), while still giving you friendly display units like degrees, minutes and seconds that match what you see on instruments and diagrams. :contentReference[oaicite:4]{index=4}
References and further reading on angle units
These references go deeper into how angle is defined in the SI, and how standard degree–radian and related conversions are set up:
- NIST Special Publication 330 — SI, Section on plane angle — explains the definition of the radian as the coherent SI unit of plane angle and shows how 360° relates to 2π radians. :contentReference[oaicite:5]{index=5}
- NIST Guide to the SI, Appendix B.9 — Angle conversion factors — lists factors for degrees, gradians, minutes, seconds, mils and revolutions in terms of radians. :contentReference[oaicite:6]{index=6}
- Radian — SI derived unit of angle — gives a concise geometric definition of the radian, with relationships to degrees, gradians and turns plus common formulas used in mathematics and physics. :contentReference[oaicite:7]{index=7}
For mission-critical engineering, surveying or navigation work, always compare quick calculator results with your organisation’s official angle tables or instrument documentation.