What this angle converter does
This tool takes an angle in one unit and instantly shows its value in the six units people use most: degrees (°), radians (rad), gradians or gons (gon), turns (full revolutions), arcminutes (’) and arcseconds (”). It is handy for trigonometry, surveying, navigation, astronomy, and any calculator or programming language that works in radians while you think in degrees. Everything runs in your browser — nothing is sent to a server.
A common case: Math.sin in JavaScript expects radians, so if you know an angle is 30 degrees you need to convert it to radians first. Or your survey plan lists slopes in gons and you want them back in degrees.
How to use the converter
- Type the amount in the Value field (it accepts decimals with a dot or comma, and negatives).
- Pick the source unit from the dropdown.
- Read the equivalents: the dark card shows degrees and the white cards show the rest. When the radian value forms a clean fraction of π (like π/4), it is noted underneath.
- Tap any card to copy the value with its symbol.
The formula and the factors
Every conversion goes through one base unit: the degree. Your value is first turned into degrees by multiplying by the source factor, then divided by the target factor. The core relationships are:
- 360° = 2π rad = 400 gon = 1 turn
- 1° = 60’ (arcminutes) = 3600” (arcseconds)
These are the exact factors, given as the value of one unit in degrees:
| Unit | Symbol | 1 unit in degrees |
|---|---|---|
| Degree | ° | 1 |
| Radian | rad | 57.2958 (= 180/π) |
| Gradian | gon | 0.9 |
| Turn | turn | 360 |
| Arcminute | ’ | 0.0166667 (= 1/60) |
| Arcsecond | ” | 0.000277778 (= 1/3600) |
The gradian (gon) splits the right angle into 100 parts instead of 90, so a full circle is 400 gon; it is common in European surveying. The turn is convenient for whole rotations: half a turn is 180° and a quarter turn is 90°.
Worked example
Let us convert 45 degrees to every unit:
- To radians: 45 × π/180 = 0.785398 rad, which is exactly π/4.
- To gradians: 45 ÷ 0.9 = 50 gon.
- To turns: 45 ÷ 360 = 0.125 turns (an eighth of a rotation).
- To arcminutes: 45 × 60 = 2,700’.
- To arcseconds: 45 × 3600 = 162,000”.
And to double-check the classic half-turn: 180° = 3.14159 rad (π) = 200 gon = 0.5 turns = 10,800’ = 648,000”.
Radians versus degrees
Degrees feel natural (a circle has 360°, a number with many divisors), but mathematics uses radians because they measure the arc in units of the radius: 2π radians cover the whole circumference. That is why sin, cos and tan in almost every programming language expect radians. The key fact is that 1 radian ≈ 57.2958°, a little over 57 degrees.
Frequently asked questions
How many degrees are in a radian?
One radian equals 180/π ≈ 57.2958°. Going the other way, one degree is π/180 ≈ 0.0174533 radians, and a right angle (90°) is π/2 ≈ 1.5708 rad.
What is a gradian or gon?
It is a unit where a right angle is 100 gon and a full turn is 400 gon. Surveyors and topographers favor it because it simplifies slope math: 1 gon = 0.9°.
What are arcminutes and arcseconds for?
They measure very small angles precisely, mainly in astronomy and geographic coordinates. A degree splits into 60 arcminutes and each arcminute into 60 arcseconds, so 1° = 3600”. The Moon, for instance, spans about 31’ across.
Can I enter negative angles or values above 360°?
Yes. The converter never clips or normalizes the input: type 720° and you see 2 turns and 4π rad; type -90° and you get the negative equivalents, preserving the direction of rotation.