Degrees and Radians Converter
Convert angles between degrees, radians, gradians and turns, with radians given exactly as a multiple of π, such as π/6 for 30°, as well as in decimals.
Result
Converted: 0.5235987756 rad
Exact multiples of π where the entry allows; decimals to ten significant figures.
- Exactly, in radians
- π/6 rad
- In every unit
- 30° = 0.5235987756 rad = 33.33333333 grad = 0.08333333333 turn
- Where it points
- Quadrant I
https://calcgallery.com/notation-conversions/degrees-radians-converter-calculator/
Unit conversions follow from exact definitions and published notation rules. Sizes follow published standards and are approximate: makers grade their own, so check their chart. Figures are rounded only for display. Read the full disclaimer
How to use the Degrees and Radians Converter
- Type the angle. Use a plain number such as 30 or 22.5, a fraction such as 1/3, or, for radians, a multiple of π such as pi/3.
- Choose the unit it is in and the unit you want.
- Read the result, the exact multiple of π, and the angle in every unit. Tick the option to also bring it within one turn.
The conversions
1 turn = 360° = 2π radians = 400 gradiansradians = degrees × π ÷ 180degrees = radians × 180 ÷ πWhy the exact form matters
Any angle that is a whole or fractional number of degrees is a fraction of π in radians: 45° is π/4, 270° is 3π/2, 22.5° is π/8. The page reduces the fraction to lowest terms, so the answer can be read at a glance and used in a formula without rounding. A decimal such as 0.7854 loses that; it is only an approximation of π/4.
Angles beyond one turn
An angle of 450° is not the same as 90°, even though it points the same way. It is a turn and a quarter, 5π/2 radians, and in rotation or wave problems the extra turn matters. The page keeps the angle as entered, and gives the angle within one turn only when asked, with the quadrant it points into.
With your numbers
- radians = degrees × π ÷ 180
Quick reference
Exact multiples of π where the entry allows; decimals to ten significant figures.
| Example | Converted |
|---|---|
| 30° | 0.5235987756 rad |
| 45° | 0.7853981634 rad |
| 90° | 1.570796327 rad |
| 270° | 4.71238898 rad |
| 450° | 7.853981634 rad |
| π/3 rad | 60° |
| 1 rad | 57.29577951° |
| 100 grad | 90° |
Common questions about the Degrees and Radians Converter
- What is 30 degrees in radians?
- Exactly π/6, which is about 0.5236. A full turn is 360° or 2π radians, so one degree is π/180 radians and 30° is 30π/180, which reduces to π/6. The exact form is usually what a maths problem wants; the decimal hides it.
- How do I convert radians to degrees?
- Multiply by 180 and divide by π. π/3 radians is 180/3 = 60°. One radian, with no π, is 180/π ≈ 57.2958°, an angle that is not a tidy number of degrees at all.
- What is a radian?
- The angle at the centre of a circle that cuts off an arc as long as the radius. A full circle’s circumference is 2π radii, so a full turn is 2π radians. Radians make calculus and physics formulas simpler, which is why they are the standard unit in mathematics.
- What are gradians?
- A unit that divides a full turn into 400, so a right angle is 100 gradians. It came from the metric reforms in France and is still found on some calculators and in surveying.