Angle Converter
Degrees, radians, gradians and turns.
Convert
Result
0.0174532925 rad
1 degree in radian
1 ° = 0.0174532925 rad
Angle Conversion Math Formula
An angle measures an amount of turning, and each unit divides the full circle differently. A degree is 1/360 of a turn and a gradian is 1/400. A radian comes from the geometry itself: it is the angle at the centre of a circle that cuts off an arc as long as the radius. A circle's circumference is 2π times its radius, so a full turn is 2π radians.
That gives the one relationship everything else comes from: 360° = 2π rad = 400 grad = 1 turn. The converter goes through degrees, so a radian picks up a factor of 180/π, which is 57.2957795… degrees per radian, and the arcminute and arcsecond are plain sixtieths.
The formula
degrees = radians × 180 ÷ πradians = degrees × π ÷ 180gradians = degrees × 10 ÷ 91° = 60′ = 3,600″
Worked example 1: A right angle, 90°, in radians
90° × (π rad ÷ 180°) = π/2 radAnswer: 1.5708 rad
The exact value is π/2. Leaving π in the answer is how you keep it exact in a maths class, and 1.5708 is the rounded decimal.
Worked example 2: One radian, in degrees
1 rad × (180° ÷ π rad)Answer: 57.2958 °
A bit under 60°. That is why a half turn of 180° needs π ≈ 3.14 radians, a little over three of them.
Worked example 3: 12.375° in arcminutes
12.375° × (60′ ÷ 1°)Answer: 742.5 ′
Split it for the clock-style form: 742.5′ is 12° and 22.5′, because 12 × 60 = 720. Then 0.5′ × 60 = 30″, so 12.375° is 12° 22′ 30″.
Worked example 4: 150 gradians, in degrees
150 grad × (0.9° ÷ 1 grad)Answer: 135 °
A right angle is 100 grad, so 150 grad is a right angle and a half: 90° + 45° = 135°.
Common mistakes to avoid
- A calculator in the wrong mode. sin(30) is 0.5 only in degree mode. In radian mode the same keys give about −0.988, because 30 radians is not 30 degrees. When a trig answer looks odd, check the mode before anything else.
- Rounding π too early. Replacing π with 3.14 causes an error of about 0.05%. Across a long chain of steps or a large angle it shows up. Keep the full π until the last line.
- Arcminutes are not minutes of time. An arcminute (′) is 1/60 of a degree and an arcsecond (″) is 1/60 of an arcminute. They share names with time units, but 90′ is 1.5°, not an hour and a half.
About angle units
Each angle unit suits a different job. Degrees are used in everyday life, geometry, surveying and navigation. Radians are the natural unit in calculus and physics, because formulas such as arc length s = rθ and the derivative of sin x only come out clean when the angle is in radians. Gradians appear in some surveying and civil engineering, where a right angle is a tidy 100 grad.
Arcminutes and arcseconds split the degree for fine measurement in astronomy, map coordinates and optics. A nautical mile was originally one arcminute of latitude, which is why latitude and nautical miles go together on a chart. A turn counts whole rotations, with 1 turn = 360°.
- Degree (°)One 360th of a full turn.
- Radian (rad)1 rad = 57.29577951 °The angle whose arc equals the radius. A full turn is 2π radians.
- Gradian (grad)1 grad = 0.9 °One 400th of a turn, so a right angle is 100 grad. Also called a gon.
- Arcminute (′)1 ′ = 0.0166666667 °One 60th of a degree.
- Arcsecond (″)1 ″ = 0.0002777778 °One 60th of an arcminute. Astronomers measure star positions in arcseconds.
- Turn (turn)1 turn = 360 °One full rotation, 360°.
Angle conversion table
| Value | ° | rad | grad | ′ |
|---|---|---|---|---|
| 1 ° | 1 | 0.0174533 | 1.11111 | 60 |
| 1 rad | 57.2958 | 1 | 63.662 | 3,437.75 |
| 1 grad | 0.9 | 0.015708 | 1 | 54 |
| 1 turn | 360 | 6.28319 | 400 | 21,600 |
| 1 ″ | 0.000277778 | 0.0000048481 | 0.000308642 | 0.0166667 |
Angle converter FAQ
How do I convert degrees to radians?
Multiply by π/180, which is about 0.0174533. 45° × π/180 = π/4 ≈ 0.7854 rad.
How do I convert radians to degrees?
Multiply by 180/π, about 57.2958. 2 rad × 57.2958 = 114.59°. For a multiple of π, just put 180 in place of π: 3π/2 rad is 3 × 180 ÷ 2 = 270°.
Why do mathematicians use radians?
They make the calculus formulas simple. The derivative of sin x is cos x only when x is in radians. In degrees there would be an extra factor of π/180 every time. Radians also give arc length directly: arc = radius × angle.
How many degrees are in a gradian?
0.9. A gradian is 1/400 of a turn, so 360° ÷ 400 = 0.9°. A 100 grad angle is therefore exactly 90°.
How do I convert decimal degrees to degrees, minutes and seconds?
Keep the whole degrees. Multiply the decimal part by 60 for minutes, keep the whole minutes, and multiply the new decimal part by 60 for seconds. 40.7128° is 40° and 0.7128 × 60 = 42.768′, so 42′ and 0.768 × 60 = 46.08″. That is 40° 42′ 46.08″.
Why are there 360 degrees in a circle?
Nobody knows for sure. The usual explanation is that the Babylonians counted in base 60, and 360 divides evenly by many numbers (it has 24 divisors), so fractions of a circle come out whole.