Angle Converter
Convert an angle between degrees, radians, gradians, arcminutes, arcseconds, milliradians, NATO mils, turns and degrees-minutes-seconds. The unit circle shows where the angle lands, with its sine, cosine and quadrant.
Last updated:
Type a number, a multiple of π such as π/4 or 2pi/3, or (with From = DMS) 45° 30′ 15″.
How to use the Angle Converter
- Type the angle: a number, a multiple of π, or degrees-minutes-seconds.
- Choose the From and To units. Pick DMS in From to enter 45 30 15 style angles.
- Read every other unit, plus the normalized angle, quadrant and trig values.
- Check the unit circle to see where the angle points.
- Copy the results or download the common-angles table.
Angle conversion formulas
The converter turns every input into degrees first, then divides by the size of the target unit in degrees.
| Unit | Full circle | 1 unit in degrees | Formula from degrees |
|---|---|---|---|
| Degree (°) | 360 | 1 | — |
| Radian (rad) | 2π ≈ 6.283185 | 57.29578 | rad = deg × π ÷ 180 |
| Gradian (grad, gon) | 400 | 0.9 | grad = deg ÷ 0.9 |
| Arcminute (′) | 21,600 | 1/60 | arcmin = deg × 60 |
| Arcsecond (″) | 1,296,000 | 1/3,600 | arcsec = deg × 3,600 |
| Milliradian (mrad) | ≈ 6,283.185 | 0.0572958 | mrad = deg × 1,000π ÷ 180 |
| NATO mil | 6,400 | 0.05625 | mil = deg × 6,400 ÷ 360 |
| Turn (revolution) | 1 | 360 | turns = deg ÷ 360 |
DMS: decimal degrees = D + M ÷ 60 + S ÷ 3,600. Going the other way, keep the whole degrees, multiply the fraction by 60 for minutes, and multiply the remaining fraction by 60 again for seconds.
Reading the unit circle
On a circle of radius 1 centered at the origin, an angle measured counterclockwise from the positive x-axis ends at the point (cos θ, sin θ). The diagram above draws that point, the cosine (horizontal) and sine (vertical) projections, and the arc for the normalized angle.
- Quadrant I (0–90°): sin and cos both positive.
- Quadrant II (90–180°): sin positive, cos negative.
- Quadrant III (180–270°): both negative, tan positive.
- Quadrant IV (270–360°): cos positive, sin negative.
Practical tips
- Calculators and spreadsheet functions such as SIN() and COS() expect radians unless they are set to degree mode.
- For rifle scopes, 1 mrad covers about 10 cm at 100 m, and 1 MOA (one arcminute) covers about 2.9 cm at 100 m (1.047 in at 100 yd).
- Small angles in astronomy are written in arcminutes and arcseconds: the full Moon is about 31′ across.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π/180. For example 45° × π/180 = π/4 ≈ 0.785398 rad. To convert radians to degrees, multiply by 180/π (about 57.2958).
What is a gradian?
A gradian (also called a gon or grad) divides a right angle into 100 parts, so a full circle is 400 gradians and 1 grad = 0.9°. Gradians are used in some surveying and civil engineering work.
What is the difference between a milliradian and a mil?
A milliradian is exactly 1/1,000 of a radian, so a circle has about 6,283.19 mrad. The NATO mil rounds this to 6,400 per circle for easy division. 1 mrad ≈ 0.0573° ≈ 3.438 arcminutes, while 1 NATO mil = 0.05625°.
How many arcminutes and arcseconds are in a degree?
One degree = 60 arcminutes and one arcminute = 60 arcseconds, so 1° = 3,600 arcseconds. In DMS form, 23.4392° is 23° 26′ 21.1″.
What is a reference angle?
It is the acute angle between the terminal side of an angle and the x-axis. For 150° it is 30°, for 225° it is 45°. Trig functions of an angle equal those of its reference angle, apart from the sign set by the quadrant.
How do I normalize an angle to 0–360°?
Add or subtract whole turns of 360° until the result is between 0 and 360. For −450°, adding 2 × 360 gives 270°. These angles are called coterminal because they point the same way.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.