📐 Angle Converter

How angle conversion works

Every angle unit is defined as a fixed fraction of a full circle, and this tool routes every conversion through radians as the common base. The exact relationships are:

• Degrees: 1° = π/180 rad ≈ 0.0174533 rad. A full circle is 360°.
• Radians: the SI unit of angle. A full circle is 2π rad ≈ 6.28319 rad.
• Gradians (gon): 1 gon = π/200 rad ≈ 0.0157080 rad. A full circle is 400 gon — a system built around decimal fractions of a right angle (100 gon = 90°).
• Turns (rev): 1 turn = 2π rad. One turn is simply one full revolution, so 0.5 turns = 180° and 0.25 turns = 90°.
• Arcminutes: 1′ = (1/60)° = π/10,800 rad ≈ 0.000290888 rad.
• Arcseconds: 1″ = (1/3600)° = π/648,000 rad ≈ 0.00000484814 rad.

To convert, the tool first works out how many radians your input represents by multiplying the value by the source unit's factor, then divides by the target unit's factor. For example, 90° becomes 90 × π/180 = π/2 rad ≈ 1.570796 rad; switching the target to gradians divides by π/200 to give exactly 100 gon. Because π cancels out of the ratio, round numbers like 90°, 180°, and 360° nearly always land on clean values in the other systems.

Worked examples

Here are a few conversions you can verify by hand:

• 180° to radians: 180 × π/180 = π rad ≈ 3.141593 rad. Half a circle is always π radians — the single most useful fact in trigonometry.
• 2.5 rad to degrees: 2.5 × (180/π) = ≈ 143.239°. One radian is about 57.2958°, so multiplying by that factor works whenever you are without a tool.
• 60° to gradians: 60 × (10/9) = 66.6667 gon. Since 400 gon = 360°, each degree equals 10/9 gon.
• 1 arcsecond to degrees: 1 ÷ 3600 = ≈ 0.000278°, or about 4.85 microradians. Tiny, but that is the precision used to aim telescopes and survey land.

A practical scenario: you are writing a small JavaScript animation that rotates a dial 270° clockwise across 2 seconds. CSS and canvas APIs accept both degrees and radians depending on the function, but `Math.sin()` and `Math.cos()` — which you likely use to compute the dial's position — require radians. Type 270 into the converter with degrees as the source and radians as the target: 270° = 4.712389 rad, which is exactly 3π/2. That single lookup turns a 20-minute debugging session into a 5-second fix.

Edge cases & limitations

• Negative and large angles: the converter accepts any real number, including negative angles (rotation the other way) and values beyond one full circle, such as 720° or 7π rad. These are perfectly valid and simply represent more than one revolution.
• Why radians look "too big": one radian is roughly 57.3°, which surprises many people. The reason is that a radian is defined so that the arc length equals the radius — so a full circle of radius 1 has circumference 2π, not 360. Small degree values therefore produce small radian values: 30° is only 0.5236 rad.
• Gradian vs. degree confusion: the gradian (unit symbol gon, sometimes called "grad") is used mainly in surveying and some European engineering fields. A calculator or programming library set to "grad" mode on a 400-scale will return wildly different trigonometric results — one common reason calculators appear "broken". This converter uses gon = 100 per right angle, matching the international standard.
• Arcminutes vs. minutes of time: an arcminute (′) is 1/60 of a degree and has nothing to do with clock minutes. It is used for angular precision in astronomy, optics, and navigation — e.g. a "20/20" acuity chart is based on a 1 arcminute detail size.
• Display rounding: results are shown with up to 6 significant decimal places, which is far beyond what any practical use needs. If you converted a huge value to a tiny unit (say, 1000 turns to arcseconds), expect up to 129,600,000″ — the converter keeps the full number rather than forcing scientific notation.