Degree to Angle Converter: Minutes, Radians & More

This converter turns a decimal degree value (like 45.5°) into the equivalent angle in degrees-minutes-seconds, radians, and gradians, and also shows the reverse conversions. You enter one angle in any of the four common units and get the corresponding values in all the others, so you can move between navigation, math, and engineering formats without redoing the arithmetic.

Formula

Every angle can be expressed in several units. The core relationship is that a full circle equals 360°, 2π radians, or 400 gradians. From that:

• Radians = degrees × (π / 180)
• Gradians = degrees × (10 / 9) (since 400 gon = 360°)
• Degrees-Minutes-Seconds: the whole part of the decimal degree is the degree; multiply the fractional part by 60 to get minutes; the fractional part of that result times 60 gives seconds.

Variables: deg = angle in decimal degrees (°), rad = angle in radians (rad), gon = angle in gradians (gon), dms = angle in degrees, minutes, and seconds (° ‘ “).

Worked Example

Using the default value of 45.5°:

1. Radians: 45.5 × (π / 180) = 45.5 × 0.0174533 ≈ 0.794125 rad.
2. Gradians: 45.5 × (10 / 9) = 50.5556 gon.
3. DMS: The whole degree part is 45. The fractional part 0.5 × 60 = 30 minutes. The fractional part of 30 is 0, so seconds = 0. Result: 45° 30′ 0″.

So 45.5° equals 0.794125 rad, 50.5556 gon, and 45° 30′ 0″.

Why Convert Between Angle Units?

Different fields use different angle conventions. Navigation and GPS coordinates are almost always given in degrees-minutes-seconds (or decimal degrees). Trigonometry in calculus and physics uses radians because they simplify derivative formulas. Surveying and some European engineering work use gradians (also called gons or grads) because a right angle is exactly 100 gon, making right-angle calculations easy. If you work with maps, CNC machines, or satellite dishes, you will likely need to switch between these units at some point.

For example, a GPS coordinate like 40° 26′ 46″ N is often stored as 40.446111° in databases. Converting between the two is essential for mapping software and geocaching. Similarly, a machinist programming a rotary axis might input angles in decimal degrees, while the controller expects radians for certain functions.

How to Read the Result

The first output, DMS, is shown as three numbers: degrees, minutes, and seconds. Minutes range from 0 to 59.999, and seconds from 0 to 59.999. If you see 45° 30′ 0″, that means 45 degrees plus 30 minutes (half a degree) plus zero seconds. The radian output is a pure number (no unit symbol is used in mathematics, though “rad” is sometimes written). Gradians are often denoted “gon” or “grad”.

When converting from DMS back to decimal degrees, the formula is: decimal degrees = degrees + (minutes / 60) + (seconds / 3600). This converter does that internally if you enter a DMS value as a decimal (e.g., 45.5 for 45°30′).

Common Mistakes and Misconceptions

• Confusing radians with degrees: A common error is typing 45 into a calculator set to radians, or vice versa. Always check the mode.
• Forgetting that minutes and seconds are base-60: 1 degree = 60 minutes, 1 minute = 60 seconds. Do not treat them as decimal fractions.
• Mixing gradians and degrees: 100 gon = 90°, so 45° is 50 gon, not 45 gon.
• Rounding too early: If you round intermediate steps, the final seconds can be off by a few units. Keep extra digits until the end.

A common misconception is that radians are just degrees multiplied by 57.3. That is true (since 180/π ≈ 57.2958), but it is safer to use the exact factor π/180.

How to Choose the Right Unit

There is no “best” unit, only the one that fits your task:

• Degrees: Everyday geometry, navigation, construction, and most school math.
• Radians: Calculus, physics, Fourier transforms, and any formula involving derivatives of trigonometric functions.
• Gradians: Land surveying in some countries, and certain military or engineering applications where right angles are common.
• DMS: GPS coordinates, astronomy (right ascension/declination), and aviation charts.

If you are unsure, degrees are the safest default for non-scientific work.

Limits of This Converter

This calculator works for any real angle, but note that angles are periodic: 45° and 405° are the same direction. The converter does not reduce angles to a 0–360° range, so you may see values like 405° in DMS. That is fine mathematically, but if you need a normalized angle, subtract 360° repeatedly until it falls in your desired range. Also, the DMS output is rounded to 4 decimal places in the seconds field, which is far more precise than any real-world measurement needs—1 second of arc is about 30 meters on the Earth’s surface.

For negative angles, the converter works as expected: -45.5° becomes -0.794125 rad and -50.5556 gon, and DMS shows -45° 30′ 0″. Some users prefer to use absolute values for DMS, but the sign is preserved here.

Frequently Asked Questions

How do I convert degrees to radians without a calculator?

Multiply the degree value by π/180. For example, 180° × π/180 = π rad, 90° × π/180 = π/2 rad. If you need a decimal, use 3.14159 for π.

What is the difference between a gradian and a degree?

A gradian (gon) divides a right angle into 100 parts, so a full circle is 400 gon. A degree divides a right angle into 90 parts, so a full circle is 360°. One degree equals 10/9 gon, or approximately 1.111 gon.

Why do some calculators have a “DEG” and “RAD” mode?

Those modes tell the calculator whether to interpret trigonometric function inputs as degrees or radians. If you enter sin(30) in DEG mode, you get 0.5; in RAD mode, you get -0.988 (since 30 radians is a different angle). Always match the mode to the unit you are using.

How do I convert DMS to decimal degrees in my head?

Divide the minutes by 60 and the seconds by 3600, then add them to the degrees. For example, 10° 30′ 15″ = 10 + 30/60 + 15/3600 = 10.50417°.

Can I use this converter for angles in a triangle?

Yes, but remember that triangle angles are usually given in degrees and sum to 180°. If you have a triangle angle in radians, convert it to degrees first to use the sum property.

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in deg | Decimal degrees | ° | 45.5 | 0
out dms | Degrees-Minutes-Seconds | ° ' " | deg | 4
out rad | Radians | rad | deg * (pi / 180) | 6
out grad | Gradians | gon | deg * (10 / 9) | 4