Why there are so many units for one thing
An angle is a fraction of a full turn, and every unit is just a different number of pieces to cut that turn into. Degrees cut it into 360, a Babylonian choice that survives because 360 divides evenly by everything from 2 to 12 except 7. Gradians cut it into 400 so that a right angle is a round 100, which is convenient for surveyors and almost nobody else. NATO mils cut it into 6400, chosen so that one mil subtends roughly one metre at a thousand metres — close enough for artillery and rangefinding. Turns cut it into one, which is the honest unit for anything rotating.
Radians are the odd one out because they are not an arbitrary division. One radian is the angle that makes the arc as long as the radius, so there are exactly 2π of them in a turn. That is an ugly number to say out loud and the reason every formula in calculus is clean: arc length becomes rθ, sector area becomes half r²θ, and the derivative of sine is cosine with no stray constants. If a formula you are copying has no degree symbol anywhere in it, it wants radians.
Degrees, minutes and seconds
The same base-60 habit that gives us hours and minutes splits a degree into 60 minutes and each minute into 60 seconds. Written 30° 15' 00", that is 30.25 degrees. Land surveys, deed descriptions and telescope coordinates still use it, and it is the notation that survives on older instruments. Converting is straightforward — divide the minutes by 60 and the seconds by 3600 and add — but the mistake to watch for is treating 30° 15' as 30.15 degrees. It is 30.25, and on a property line a tenth of a degree is a few inches at the far corner.
Setting an angle without a protractor
Two methods cover almost everything, and both trade a protractor for a tape measure, which is the better instrument anyway.
Offset from a baseline. Measure a known distance along the reference line — call it the run — then measure perpendicular from that point by run × tan(angle). Strike a line from the corner through the new mark. A 30 degree angle over a 24 inch run needs a 13.86 inch offset. The longer the run, the less a marking error matters: a sixteenth of an inch off at 12 inches is a third of a degree, but the same error at 48 inches is under a tenth.
Chord across an arc. Swing the same radius from the corner across both legs, then set the second leg so the two arc marks are 2R · sin(half the angle) apart. For 30 degrees at a 24 inch radius that is 12.42 inches. This one works past 90 degrees, where the offset method needs an impractically long perpendicular, and it does not require anything to be square to start with.
The one conversion that catches people
Percent grade is not degrees and never approaches them. A 100 percent grade is 45 degrees, because it rises as much as it runs. The two track loosely at shallow angles — 5 percent is 2.86 degrees, 10 percent is 5.71 — and then diverge without limit, since the tangent runs to infinity at 90 while the angle stays finite. Anyone quoting a 45 percent hill and a 45 degree hill as the same thing is out by nearly a factor of two. That is also why this page shows a dash instead of a grade at 90 degrees: there is no horizontal run left to divide by, and the answer is not a large number, it is no number.
Questions people ask
How many degrees is one radian?
About 57.2958 degrees, which is 180 divided by pi. Going the other way, one degree is 0.0174533 radians. The conversions worth committing to memory are the landmarks rather than the constant: pi radians is a half turn at 180 degrees, pi over 2 is a right angle, pi over 4 is 45 degrees, and pi over 6 is 30. Most trigonometry written for programming works in radians by default, so a formula that returns nonsense when you feed it degrees is usually not wrong, just expecting the other unit.
What is a gradian and does anyone use it?
A gradian, also called a gon, is one four-hundredth of a turn, so a right angle is exactly 100 gradians and a full circle is 400. It was a metric-era attempt to decimalise the circle. It survives in some European surveying and civil engineering practice and on the scientific calculators that carry a DEG/RAD/GRAD switch, which is where most people meet it — usually by knocking the switch into GRAD by accident and wondering why every answer is out by about eleven percent.
Can I use this for a roof pitch or a drainage fall?
You can read the conversion here — the reference table gives the rise in 12 for any angle — but two other pages on this site are built for those jobs specifically. The roof pitch calculator works in the rise-in-12 notation throughout and adds rafter multipliers. The slope and grade calculator handles percent grade, 1-in-X ratios and fall per foot, which is the vocabulary of drainage and ramps. This page is aimed at the case where you have an angle in one unit and need it in another, or you need to mark it on material.
Why does the tangent go undefined at 90 degrees?
Because the tangent is rise divided by run, and at 90 degrees the run is zero. Dividing by zero has no answer, and approaching it the tangent grows without bound: 89 degrees is a grade of 5,729 percent, 89.9 is 57,290 percent. Nothing is broken when the calculator declines to print a number there. It also means percent grade is a poor way to describe anything steep — above about 30 degrees the figures become large and hard to compare, which is why steep ground gets quoted in degrees and gentle ground in percent.