Degrees to Radians Calculator
Convert degrees, radians, gradians and turns, with exact multiples of pi.
About
Degrees to Radians Calculator
Degrees and radians measure the same thing and answer to different audiences. Degrees suit navigation and construction; radians suit anything involving calculus. Converting between them is one multiplication, and knowing which to use matters more than the arithmetic.
The formulas
radians = degrees × π ÷ 180
degrees = radians × 180 ÷ π
Both come from the same fact: a half turn is 180 degrees and also π radians, so the two are interchangeable as a conversion factor.
Worked examples
- 45° = 45 × π ÷ 180 = π/4 = 0.785398 radians
- 90° = π/2 = 1.570796 radians
- 2 radians = 2 × 180 ÷ π = 114.592°
Answers in terms of π are exact; decimals are rounded. Keeping the π form through a calculation and converting once at the end avoids accumulating rounding error, which is why textbooks insist on it.
The angles worth memorising
| Degrees | Radians | Decimal |
|---|---|---|
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 180° | π | 3.1416 |
| 270° | 3π/2 | 4.7124 |
| 360° | 2π | 6.2832 |
What a radian actually is
One radian is the angle that cuts an arc equal in length to the radius. Take the radius, bend it around the circle, and the angle it subtends is one radian, about 57.296 degrees.
Since the circumference is 2πr, a full circle holds 2π radians, roughly 6.283. The number is awkward and the definition is not arbitrary, which is the trade the unit makes.
Why radians are used in calculus
Two payoffs follow from that definition.
Arc length and sector area become clean. Arc length is rθ and sector area is ½r²θ, with no conversion factor. In degrees they need π/180 sprinkled through them.
More importantly, the derivative of sin x is cos x exactly, but only in radians. In degrees the derivative picks up a factor of π/180, which then propagates through every result built on it. Every Taylor series, every differential equation and every complex exponential in mathematics assumes radians, which is why programming languages take radians in their trigonometric functions and why passing degrees produces silently wrong answers.
Why a circle has 360 degrees
The number comes from Babylonian astronomy and its base-60 counting system, with a plausible link to a year of roughly 360 days. It survived because it is unusually convenient: 360 divides evenly by 24 different numbers, including 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 60, 72, 90, 120 and 180.
That divisibility is why a circle can be cut into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths and still land on whole degrees. A base-100 system would fail at thirds, which matters in practice.
Gradians and turns
A gradian divides a right angle into 100 rather than 90, so a full circle holds 400. It came out of the French metric reforms and survives mainly in surveying, where a gradian conveniently equals one centesimal minute of arc. Most scientific calculators still carry a GRAD mode, and it is the setting that produces wrong answers when someone leaves it on by accident.
A turn is one full revolution, which makes it the most intuitive unit of the four. Half a turn is 180 degrees. Rotational speeds are quoted in turns per minute, usually written as rpm.
Degrees, minutes and seconds
Angles in navigation and astronomy subdivide the degree the way an hour subdivides: 60 minutes to a degree, 60 seconds to a minute. This is the same Babylonian base-60 inheritance.
So 45.5075° is 45° 30' 27". Latitude and longitude are still commonly written this way, though decimal degrees have largely taken over in software because they are easier to compute with. One second of latitude is about 31 metres, which sets the precision the notation implies.
The small angle approximation
For small angles measured in radians, sinθ and tanθ are both close to θ itself. This holds only in radians, and it is one of the clearest demonstrations of why the unit exists.
| Angle | Radians | sinθ | Error |
|---|---|---|---|
| 1° | 0.01745 | 0.01745 | 0.005% |
| 5° | 0.08727 | 0.08716 | 0.13% |
| 10° | 0.17453 | 0.17365 | 0.51% |
| 20° | 0.34907 | 0.34202 | 2.06% |
Under 10 degrees the substitution costs half a percent, which is inside the tolerance of most physical measurements. It is what makes the pendulum equation solvable, and it underpins the optics of small-angle deflection and the way astronomers convert angular sizes into distances.
Angular velocity
Rotation rates convert the same way. To go from revolutions per minute to radians per second, multiply by 2π and divide by 60.
An engine at 3,000 rpm is turning at 3,000 × 2π ÷ 60 = 314.16 rad/s. A vinyl record at 33.3 rpm turns at 3.49 rad/s. The Earth, completing one rotation in 86,164 seconds, turns at 7.29 × 10-5 rad/s.
Radians per second is the unit that makes the physics work, since the speed of a point on a rotating body is simply rω, with no conversion factor. A blade tip 0.3 m from the centre of a 3,000 rpm motor is moving at 0.3 × 314.16 = 94.2 m/s.
Bearings and standard position
Mathematics and navigation measure angles from different places and in different directions, which catches people moving between the two.
Standard mathematical position measures anticlockwise from the positive x axis, so 90° points up. A compass bearing measures clockwise from north, so 90° points east. Converting between them means both reflecting and rotating, and the mapping is bearing = 90 - standard angle, taken modulo 360.
Anyone writing software that mixes a mapping library with a trigonometric one meets this immediately, usually as a result rotated a quarter turn from where it should be.
Where each unit belongs
Degrees suit anything measured, drawn or communicated between people. Construction drawings, compass bearings, camera fields of view and protractors all use them, because whole degrees are fine enough for the job and the numbers stay friendly.
Radians suit anything computed. Any formula involving a derivative, an integral, a series expansion or a complex exponential assumes them, and substituting degrees breaks the result rather than merely rescaling it. The practical rule is to work in radians and convert to degrees only for display, which is what most engineering software does internally.
Negative and coterminal angles
A negative angle simply rotates the other way, so -90° is a quarter turn clockwise and lands in the same place as +270°. Angles that share a final position are called coterminal, and they differ by whole multiples of 360° or 2π.
Normalising an angle into the 0 to 360 range is often the first step in a calculation, since 750° and 30° behave identically in every trigonometric function. This calculator shows the normalised value alongside the raw one, because the two answer different questions: the raw figure records how far something turned, and the normalised figure records where it ended up.
Common mistakes
Leaving a calculator in the wrong mode. sin(30) is 0.5 in degrees and -0.988 in radians. Check the mode indicator before trusting a trigonometric answer.
Passing degrees to a programming language. Math.sin and its equivalents take radians in essentially every language. Convert first.
Rounding π too early. Keep the exact form through the working and convert once at the end.
Confusing arc minutes with time minutes. They share a name and a base and measure different quantities.
Common questions
Frequently asked questions
Multiply the degrees by pi and divide by 180. So 45 degrees becomes 45 times pi over 180, which simplifies to pi over 4, or about 0.7854 radians. Keeping the answer in terms of pi is exact, while the decimal is rounded.
The angle that cuts an arc equal in length to the radius of the circle. Bending the radius around the circumference gives one radian, about 57.296 degrees. Since the circumference is 2 pi r, a full circle contains 2 pi radians.
Because the formulas come out clean. Arc length is r times theta and sector area is half r squared theta with no conversion factor, and the derivative of sin x equals cos x exactly. In degrees that derivative carries a factor of pi over 180 that then spreads through every result.
About 57.2958 degrees, from 180 divided by pi. It is not a round number in degrees, which is the price of defining the unit from the geometry of the circle rather than from a convenient count.
The convention comes from Babylonian astronomy and its base-60 counting, with a likely link to a year of about 360 days. It survived because 360 divides evenly by 24 different numbers, so halves, thirds, quarters, fifths and sixths all land on whole degrees.
A unit dividing a right angle into 100 parts, so a full circle holds 400 gradians. It came from the French metric reforms and survives mainly in surveying. Scientific calculators keep a GRAD mode, which is a common source of wrong trigonometric answers when left on accidentally.
Radians, in essentially every language. Math.sin, Math.cos and their equivalents all expect radians, so degrees must be converted first. Passing degrees produces a valid number that is silently wrong, which makes it a hard bug to spot.
Subdivisions of a degree, with 60 minutes to a degree and 60 seconds to a minute, following the same base-60 pattern as time. They are used in navigation and astronomy, where one second of latitude is about 31 metres on the ground.