Convert degrees, radians, gradians and degrees-minutes-seconds, with the sine, cosine and tangent of the angle.
A full turn can be written four ways, and each field of work picked a different one. Degrees split the circle into 360 parts, an inheritance from Babylonian astronomy that survives because 360 divides evenly by so many numbers. Radians measure the angle by the arc it cuts on a circle of radius 1, which makes a full turn 2π and is the only unit in which the calculus of sines and cosines works cleanly — it is also what programming languages expect. Gradians split the circle into 400, so a right angle is a round 100, a convenience for surveying. Degrees, minutes and seconds subdivide the degree in sixtieths, the notation of maps, GPS coordinates and telescopes.
The converter takes whatever you type, turns it into decimal degrees first, and derives everything else from there: radians as ° × π ÷ 180, gradians as ° × 10 ÷ 9, and the sexagesimal notation by splitting the whole degrees, then the minutes, then the seconds — with the carry handled, so a value that rounds to 60 seconds becomes one more minute instead of an impossible 0°59′60″. When the angle is a simple multiple of π it is also written that way (180° as π rad, 45° as π/4 rad), which is usually the form a textbook wants.
The three trigonometric functions are shown for the same angle to six decimals. The tangent is the one with a catch: it is the sine divided by the cosine, and the cosine is zero at 90°, 270° and every 180° after that, so the tangent has no value there — the tool prints "undefined" instead of the enormous number floating-point arithmetic would otherwise produce. Everything is computed with the browser own Math functions, on your device, and nothing you type is sent anywhere.
radians = degrees × π ÷ 180 · degrees = radians × 180 ÷ π · gradians = degrees × 10 ÷ 9 · degrees = gradians × 0.9. Sexagesimal: decimal degrees = D + M ÷ 60 + S ÷ 3600. One full turn = 360° = 2π rad = 400 gon.
Multiply by π and divide by 180. So 180° = π ≈ 3.141593 rad, 90° = π/2 ≈ 1.570796 rad and 45° = π/4 ≈ 0.785398 rad. To go the other way, multiply the radians by 180 and divide by π.
About 57.295780°. It is the angle whose arc has the same length as the radius of the circle, which is why 2π radians (about 6.283185) complete a full turn.
A unit that divides the full turn into 400 parts instead of 360, so a right angle is exactly 100 gon. It is used mostly in surveying and civil engineering, and appears on calculators as GRAD.
It is 45 degrees, 30 minutes and 0 seconds — minutes are sixtieths of a degree and seconds are sixtieths of a minute. In decimal degrees that is 45 + 30 ÷ 60 = 45.5°.
Because the tangent is sine divided by cosine, and the cosine of 90° is zero. The division has no result, so the tool shows "undefined" — the same happens at 270° and at every angle 180° apart from those.
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