Angle Converter
Ninety degrees is π/2 radians, 100 gradians and a quarter turn. Radians are the one that matters, because every trigonometric function in every programming language expects them.
Converted
1.570796radians
- degrees
- 90.000000
- radians
- 1.570796
- gradians
- 100.000000
- turns
- 0.250000
- Conversions use the defined factors rather than the rounded ones people memorise. An inch is exactly 25.4 millimetres by international agreement, not 25 and a bit, so these will not drift.
- Every unit in the family is shown, not just the one you asked for, because the next question is usually a different one.
About angle conversion
The degree is arbitrary and ancient. Three hundred and sixty of them to a circle, inherited from Babylonian astronomy, kept because 360 divides evenly by an unusual number of things — 2, 3, 4, 5, 6, 8, 9, 10, 12 and more. That divisibility is why it survived and why fractions of a circle come out as whole numbers.
The radian is not arbitrary. One radian is the angle at which the arc length equals the radius, which makes it a property of circles rather than a human choice. A full circle is 2π radians, which is about 6.283 — an awkward number that buys enormous simplification everywhere else.
That simplification is why every programming language uses radians. PHP's sin(), JavaScript's Math.cos(), Python's math.tan() — all of them take radians, and feeding them degrees is one of the most common bugs in graphics and geometry code. The symptom is an answer that looks almost right and is not.
Gradians divide the circle into 400, so a right angle is exactly 100. They were a French decimal reform and survive mainly in surveying. Turns are simply fractions of a full rotation, and are the most intuitive unit for anything that actually spins.
What it works out
- Degrees, radians, gradians and turns
- Every unit shown at once
- The unit your code actually needs
The formula
Radians = Degrees × π ÷ 180 Degrees = Radians × 180 ÷ π
Ninety degrees times π over 180 is π/2, which is 1.5708 radians.
The 180 is there because a half turn is π radians and 180 degrees — the two units agreeing at the halfway point is the whole conversion.
The values worth recognising: π/6 is 30 degrees, π/4 is 45, π/3 is 60, π/2 is 90, π is 180 and 2π is a full circle. Anything expressed as a fraction of π is almost certainly radians.
Gradians divide the circle into 400, so 90 degrees is exactly 100 gradians and a full circle is 400. The tidiness was the point — it was a decimal reform, contemporary with the metric system — and it never displaced degrees outside surveying.
A turn is the most direct unit of all: 90 degrees is 0.25 turns. For anything rotating, it is the number that means something physically.
- Degree
- A 360th of a circle. Arbitrary, ancient, and highly divisible.
- Radian
- The angle where arc length equals radius. A property of circles, not a choice.
- Gradian
- A 400th of a circle. A right angle is exactly 100.
- Turn
- A full rotation. The most direct unit for anything that spins.
A worked example
Ninety degrees in radians.
That works out to 1.570796 radians.
- degrees
- 90.000000
- radians
- 1.570796
- gradians
- 100.000000
- turns
- 0.250000
Questions
How do I convert degrees to radians?
Multiply by π and divide by 180. Ninety degrees is π/2, which is 1.5708 radians.
Why do programming languages use radians?
Because calculus and the underlying maths are far simpler in radians — the derivative of sine is cosine only if the angle is in radians. Every standard library follows, and feeding degrees to sin() is a very common bug.
What is a radian, exactly?
The angle at which the arc length equals the radius. It is defined by the circle itself rather than chosen, which is why it turns up naturally throughout mathematics.
Why are there 360 degrees in a circle?
Babylonian astronomy, and it stuck because 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10 and 12. That divisibility means most useful fractions of a circle are whole numbers.
What is a gradian?
A 400th of a circle, so a right angle is exactly 100. It was a French decimal reform alongside the metric system and survives mainly in surveying.
Which radian values should I recognise?
π/6 is 30°, π/4 is 45°, π/3 is 60°, π/2 is 90°, π is 180° and 2π is a full circle. Anything written as a fraction of π is radians.