90° in radians
- Value
- 90 °
- From
- degree (°)
- To
- radian (rad)
- Converted value
- 1.5707963268 rad
- Exactness
- Exact definitions, involves π
Checked against: 90 × π/180 = π/2 = 1.57079632679489661923…
Angle converter
Convert degrees to radians and radians to degrees, plus gradians, arcminutes, arcseconds, turns and milliradians, with π carried to 50 digits.
By AdminDrafted with AI writing toolsUpdated Checked examples: 5
Converted value
rad
Converted value: 1.57079632679 rad
- Conversion
- 90° ≈ 1.570796 rad
- Conversion factor
- 0.0174532925199
- Exactness
- Exact definitions, involves π
90° is about 1.570796 rad. One degree is about 0.01745329 rad.
The angle
90° in every Angle unit
| Unit | Value | Definition |
|---|---|---|
| radian (rad) | 1.570796 | Base unit |
| degree (°) | 90 | 1° = π/180 rad, exact |
| gradian (gon) | 100 | 1 gon = π/200 rad, exact |
| arcminute (′) | 5,400 | 1′ = π/10,800 rad, exact |
| arcsecond (″) | 324,000 | 1″ = π/648,000 rad, exact |
| turn | 0.25 | 1 turn = 2π rad, exact |
| milliradian (mrad) | 1,570.796 | 1 mrad = 0.001 rad, exact |
How it's calculated S
Convert degree to radian
1° = π/180 rad, exact
The units are defined exactly in terms of π, so the result is irrational; it is shown to 12 significant figures.
About the angle converter
The radian is the SI unit of angle: the angle whose arc equals the radius, so a full turn is 2π rad. Degrees divide the turn into 360 and gradians (gon) into 400, and arcminutes and arcseconds are 1/60 and 1/3,600 of a degree. Converting degrees to radians multiplies by π/180; irrational results are shown to 12 significant figures.
The default converts a right angle: 90° = π/2 = 1.57079632679 rad. Programmers need this because the trigonometric functions of most languages take radians, and surveyors and astronomers move between degrees, gradians and arcseconds.
Conversions among degrees, arcminutes, arcseconds, gradians and turns never involve π and are exact: 1 turn = 360° = 400 gon. The milliradian here is the true 0.001 rad, about 6,283.2 to a turn.
How to convert degrees to radians
Multiply degrees by π/180. For 75°: 75 × π/180 = 5π/12 ≈ 1.308996939 rad, and 30° = π/6 ≈ 0.523598775598 rad. To go back, multiply radians by 180/π, so 1 rad ≈ 57.2957795131°. Use radians whenever a formula, calculator mode or programming function expects them.
Write the answer as a fraction of π
Exact work keeps π in the answer. To get it:
- Put the degrees over 180: 75° gives 75/180.
- Reduce the fraction: 75/180 = 5/12.
- Attach π: 5π/12 rad.
The same steps turn 135° into 135/180 = 3/4, so 135° = 3π/4 rad, about 2.35619449019 rad. Use the decimal only when you need a number to calculate with.
Degrees to radians chart
| Degrees | Exact radians | Decimal radians |
|---|---|---|
| 0 | 0 | 0 |
| 30 | π/6 | 0.523598775598 |
| 45 | π/4 | 0.785398163397 |
| 75 | 5π/12 | 1.308996939 |
| 90 | π/2 | 1.57079632679 |
| 180 | π | 3.14159265359 |
| 360 | 2π | 6.28318530718 |
| -90 | −π/2 | -1.57079632679 |
A negative angle keeps its sign: −90° is −π/2 rad. Angles past a full turn convert the same way, so 720° is 4π rad, about 12.5663706144 rad.
How to convert radians to degrees
Multiply radians by 180/π. When the angle is a fraction of π, the π cancels: π/3 rad × 180/π = 60°. For a plain number, use 180/π ≈ 57.2957795131, so 2 rad ≈ 114.591559026°.
In degrees, minutes and seconds, one radian is about 57° 17′ 44.8″. A radian is the angle whose arc equals the circle's radius. A full turn holds 2π ≈ 6.28 of them, so each is a little less than a sixth of 360°.
Why formulas use radians
With the angle in radians, arc length is radius times angle: on a circle of radius 2 m, an angle of 1.5 rad spans 3 m of arc. The same formula in degrees needs an extra factor of π/180.
Calculus results such as "the derivative of sin x is cos x" also hold only in radians, and the trigonometric functions of most programming languages take radians.
Gradians, arcminutes, turns and milliradians
One turn is 360°, 400 gradians or 2π rad, and one degree is 60 arcminutes or 3,600 arcseconds. Conversions among these units never involve π, so the results are exact.
| Input | Output unit | Converted value |
|---|---|---|
| 1 turn | degree | 360 |
| 1 turn | gradian | 400 |
| 1 degree | arcminute | 60 |
| 1 degree | arcsecond | 3600 |
| 1 gradian | degree | 0.9 |
| 1000 milliradians | radian | 1 |
The milliradian is exactly 0.001 rad, about 6,283.2 to a turn. A military mil of 1/6,400 turn is a different, slightly smaller unit.
The SI Brochure, Table 8 gives the degree, arcminute and arcsecond in radians, and NIST's conversion tables list further angle factors.
Common mistakes
- Using the wrong calculator mode. sin 30 is 0.5 in degree mode. In radian mode the same keys return −0.988031624093, the sine of 30 radians.
- Passing degrees to code. In JavaScript,
Math.sin(90)returns about 0.894, because it reads 90 as radians. Convert first:Math.sin(90 * Math.PI / 180)returns 1. - Reading arcminutes as a decimal. 45° 30′ is 45.5°, not 45.30°. The degrees, minutes and seconds converter handles that notation.
Related tools
For sine, cosine and tangent with a chosen angle unit, use the trigonometric functions calculator. For the sides and angles of a right triangle, use the right-triangle calculator.
Worked examples
1 turn in degrees
- Value
- 1 turn
- From
- turn
- To
- degree (°)
- Converted value
- 360 °
- Exactness
- Exact
Checked against: 1 turn = 2π rad = 360° (π cancels)
1° in arcseconds
- Value
- 1 °
- From
- degree (°)
- To
- arcsecond (″)
- Converted value
- 3,600 ″
Checked against: 1° = 60′ = 3600″ (SI Brochure Table 8)
1 radian in degrees
- Value
- 1 rad
- From
- radian (rad)
- To
- degree (°)
- Converted value
- 57.2957795131 °
Checked against: 180/π = 57.29577951308232087…
Questions
How do I convert degrees to radians?
Multiply by π/180, about 0.0174533. 90° is π/2 = 1.5708 rad, 180° is π = 3.14159 rad and 45° is π/4 = 0.785398 rad. To go back, multiply radians by 180/π, so 1 rad is 57.2958°. JavaScript's Math.sin and Python's math.sin both expect radians.
How many degrees are in a radian?
57.2957795°, which is 180/π. A radian is the angle at the centre of a circle cut off by an arc as long as the radius; a full circumference is 2π radii, so a full turn is 2π rad = 360°. The SI Brochure defines the radian as 1 m/m, a coherent SI unit.
What is a gradian?
One 400th of a full turn, so a right angle is exactly 100 gradians (symbol gon, also called grads). One gon is 0.9° or 54 arcminutes. It is used mainly in surveying, where a quarter circle of 100 keeps slopes and bearings decimal, and calculators label the mode GRAD.
Is a mil the same as a milliradian?
Not always. A true milliradian is 0.001 rad, about 6,283.2 per turn, and that is the unit used here. NATO compasses and artillery use a mil of 1/6,400 turn, 0.98175 mrad, chosen so the circle divides evenly. Rifle scopes marked MRAD use the true 0.001 rad, where 1 mrad spans 10 cm at 100 m.
How many arcseconds are in a degree?
3,600: a degree is 60 arcminutes (′) and an arcminute is 60 arcseconds (″), as listed in Table 8 of the SI Brochure. One arcsecond of latitude is about 30.9 m on the ground, because the nautical mile, 1,852 m, was based on one arcminute of latitude.
About this calculator
Sources
Checked against references
This calculator includes 5 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.