Angle Converter
Convert between degrees, radians, gradians, arcminutes, and arcseconds with formula display.
Results
Converted Value
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Formula
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All Conversions
What is Angle Converter?
An angle measures the rotation between two rays that share a common endpoint, or the opening between two intersecting lines. Because so many disciplines describe rotation differently, a single measurement is expressed in several units: degrees in everyday use and education, radians in mathematics, physics, and programming, gradians in surveying and civil engineering, and arcminutes and arcseconds where fine precision is required, such as astronomy and navigation.
This converter supports all five of these units with a live result, a step-by-step formula, and a reference panel that shows the value in every other unit at once. Every conversion runs on exact mathematical relationships — π radians = 180°, 90° = 100 gradians, 1° = 60 arcminutes = 3600 arcseconds — so the results are mathematically precise rather than rounded through intermediate steps.
Conversions are computed by first normalising the input to degrees, then scaling out to the target unit. This two-step approach keeps every unit pair consistent and explains why the formula panel sometimes shows an intermediate degree value.
Key properties
- Five units — degrees, radians, gradians, arcminutes, and arcseconds.
- Any-to-any conversion — choose any source and target unit combination.
- Live formula — see the exact multiplication or division used.
- All-units panel — one glance shows every unit value simultaneously.
- Runs in the browser — no data is sent to a server.
Common Angle Conversions
These exact relationships are used by the converter and are handy to memorize for quick mental math:
| Degrees | Radians | Gradians | Arcminutes |
|---|---|---|---|
| 0° | 0 rad | 0 grad | 0′ |
| 30° | π/6 rad | 33.33 grad | 1,800′ |
| 45° | π/4 rad | 50 grad | 2,700′ |
| 90° | π/2 rad | 100 grad | 5,400′ |
| 180° | π rad | 200 grad | 10,800′ |
| 360° | 2π rad | 400 grad | 21,600′ |
Angle units reference
Each unit divides the full circle differently. The table below shows the symbol, how many fit in a complete turn, and the relationship used in every calculation.
| Unit | Symbol | Full circle | Relation to degrees |
|---|---|---|---|
| Degrees | ° | 360 | 1° = 1° (base) |
| Radians | rad | 2π ≈ 6.2832 | 1 rad = 180/π ≈ 57.2958° |
| Gradians | grad | 400 | 1 grad = 0.9° |
| Arcminutes | ′ | 21,600 | 1′ = 1/60° ≈ 0.016667° |
| Arcseconds | ″ | 1,296,000 | 1″ = 1/3600° ≈ 0.00027778° |
Arcminutes and arcseconds are often written in sexagesimal form, such as 30°15′20″ for 30.2555°. The converter works with decimal degrees, so convert such a value to decimal first if needed.
Worked examples
Each example below follows the same normalise-to-degrees-then-scale method the tool uses, so you can reproduce the result by hand.
- 90° to radians: 90 × π/180 = π/2 ≈ 1.5707963 rad (one right angle).
- π radians to degrees: π × 180/π = 180° (a straight angle).
- 200 gradians to degrees: 200 × 0.9 = 180°, so gradians map to degrees by the fixed factor 0.9.
- 45° to arcminutes: 45 × 60 = 2700′.
- 1° to arcseconds: 1 × 3600 = 3600″, so a single degree is a large number of arcseconds.
- 1 radian to gradians: normalise to degrees (1 × 180/π ≈ 57.29578°) then divide by 0.9 to get ≈ 63.66198 grad.
How to Use
- Enter Value — Type the numeric angle value you want to convert.
- Select Units — Choose the source unit (From) and target unit (To) from the dropdowns.
- View Result — The converted value appears instantly along with the conversion formula.
- Swap Units — Click the Swap button to reverse the conversion direction.
- Use Examples — Click "90°" to see a right angle conversion, or "π rad" for pi radians.
Frequently Asked Questions
What is the difference between degrees and radians?
A degree is 1/360 of a full circle and is the everyday unit for measuring angles. A radian is the angle whose arc length equals the radius; there are 2π radians in 360°, so π radians equals 180°. Radians are preferred in calculus, physics, and most programming languages because they simplify trigonometric formulas.
How do I convert degrees to radians?
Multiply the number of degrees by π divided by 180 (π/180 ≈ 0.0174533). For example, 90° × π/180 = π/2 ≈ 1.5707963 rad. The converter shows this exact multiplication in the formula panel.
How do I convert radians to degrees?
Multiply the number of radians by 180 divided by π (180/π ≈ 57.29578). For example, 1 rad × 180/π ≈ 57.29578°. Because the tool normalises to degrees first, a radian result may show the intermediate degree value in the formula.
What are gradians used for?
Gradians, also called gons, divide a right angle into 100 units so a full circle is exactly 400 gradians. They are used mainly in surveying, civil engineering, and land measurement in some European countries, where the decimal-friendly quarter circle of exactly 100 gradians simplifies calculations.
When should I use arcminutes and arcseconds?
Arcminutes and arcseconds subdivide degrees for high-precision work: 1° = 60′ and 1′ = 60″. They appear in astronomy for star positions, in navigation for GPS coordinates, in optics for lens fields of view, and in surveying for property boundaries.
How many arcseconds are in a degree?
One degree contains 60 arcminutes, and each arcminute contains 60 arcseconds, so 1° = 60 × 60 = 3600 arcseconds. A full circle therefore holds 1,296,000 arcseconds.
Can I convert negative angles or values above 360 degrees?
Yes. The converter applies the same linear formulas to any numeric value, including negatives and values greater than a full turn. Negative angles indicate rotation in the opposite direction, and values above 360° represent more than one full rotation; neither is wrapped or normalised.
Is my data stored anywhere?
No. All conversions happen instantly in your browser and your input never leaves your device.