Convert Turn to Radian
1 turn = 6.283185 radian (rad). For example, 10 turn = 62.83185 radian and 100 turn = 628.3185 radian. Type any value below to convert turn to radian both ways — it runs entirely in your browser.
Quick answer: Turn to Radian
1 Turn = 6.283185 Radian (turn → rad). To convert turn to radian, multiply by 6.283185. To convert radian to turn, multiply by 0.1591549. The converter at the top does this instantly and both ways; the sections below give the formula, worked examples, a full conversion chart and answers to the most common turn-to-radian questions.
How to convert Turn to Radian
To convert turn to radian, multiply the number of turn by the conversion factor 6.283185, because 1 turn = 6.283185 rad. The conversion is linear, so the same factor works for any value — whole numbers, decimals or fractions. For the reverse direction, 1 rad = 0.1591549 turn, so you divide by 6.283185 (or multiply by 0.1591549) to turn radian back into turn.
rad = turn × 6.283185 ·
turn = rad × 0.1591549
- Write down the value you want to convert, in turn.
- Multiply it by 6.283185.
- The result is the same area/length/quantity expressed in radian.
Turn to Radian — worked examples
Each row shows the exact arithmetic so you can follow the conversion step by step. For example, 5 turn is 31.41593 radian because 5 turn × 6.283185 = 31.41593 rad.
| Turn | Calculation | Radian |
|---|---|---|
| 1 turn | 1 turn × 6.283185 = 6.283185 rad | 6.283185 rad |
| 2 turn | 2 turn × 6.283185 = 12.56637 rad | 12.56637 rad |
| 3 turn | 3 turn × 6.283185 = 18.84956 rad | 18.84956 rad |
| 5 turn | 5 turn × 6.283185 = 31.41593 rad | 31.41593 rad |
| 7 turn | 7 turn × 6.283185 = 43.9823 rad | 43.9823 rad |
| 10 turn | 10 turn × 6.283185 = 62.83185 rad | 62.83185 rad |
| 12 turn | 12 turn × 6.283185 = 75.39822 rad | 75.39822 rad |
| 15 turn | 15 turn × 6.283185 = 94.24778 rad | 94.24778 rad |
| 20 turn | 20 turn × 6.283185 = 125.6637 rad | 125.6637 rad |
| 25 turn | 25 turn × 6.283185 = 157.0796 rad | 157.0796 rad |
| 50 turn | 50 turn × 6.283185 = 314.1593 rad | 314.1593 rad |
| 100 turn | 100 turn × 6.283185 = 628.3185 rad | 628.3185 rad |
Common Turn to Radian conversions
These are the turn-to-radian values people look up most often. Every figure is computed from the exact factor, so you can rely on them for quick reference:
- 1 turn = 6.283185 radian
- 2 turn = 12.56637 radian
- 2.5 turn = 15.70796 radian
- 3 turn = 18.84956 radian
- 4 turn = 25.13274 radian
- 5 turn = 31.41593 radian
- 6 turn = 37.69911 radian
- 7 turn = 43.9823 radian
- 8 turn = 50.26548 radian
- 9 turn = 56.54867 radian
- 10 turn = 62.83185 radian
- 12 turn = 75.39822 radian
- 15 turn = 94.24778 radian
- 20 turn = 125.6637 radian
- 25 turn = 157.0796 radian
- 30 turn = 188.4956 radian
- 50 turn = 314.1593 radian
- 75 turn = 471.2389 radian
- 100 turn = 628.3185 radian
- 200 turn = 1,256.637 radian
- 500 turn = 3,141.593 radian
- 1000 turn = 6,283.185 radian
Turn to Radian conversion chart (both directions)
The chart below converts turn to radian and radian to turn side by side, so it works whichever way you need. 1 turn = 6.283185 radian, and 1 radian = 0.1591549 turn.
| Turn → Radian | Radian → Turn |
|---|---|
| 0.25 turn = 1.570796 rad | 0.25 rad = 0.03978874 turn |
| 0.5 turn = 3.141593 rad | 0.5 rad = 0.07957747 turn |
| 1 turn = 6.283185 rad | 1 rad = 0.1591549 turn |
| 2 turn = 12.56637 rad | 2 rad = 0.3183099 turn |
| 3 turn = 18.84956 rad | 3 rad = 0.4774648 turn |
| 4 turn = 25.13274 rad | 4 rad = 0.6366198 turn |
| 5 turn = 31.41593 rad | 5 rad = 0.7957747 turn |
| 6 turn = 37.69911 rad | 6 rad = 0.9549297 turn |
| 7 turn = 43.9823 rad | 7 rad = 1.114085 turn |
| 8 turn = 50.26548 rad | 8 rad = 1.27324 turn |
| 9 turn = 56.54867 rad | 9 rad = 1.432394 turn |
| 10 turn = 62.83185 rad | 10 rad = 1.591549 turn |
| 12 turn = 75.39822 rad | 12 rad = 1.909859 turn |
| 15 turn = 94.24778 rad | 15 rad = 2.387324 turn |
| 20 turn = 125.6637 rad | 20 rad = 3.183099 turn |
| 25 turn = 157.0796 rad | 25 rad = 3.978874 turn |
| 30 turn = 188.4956 rad | 30 rad = 4.774648 turn |
| 40 turn = 251.3274 rad | 40 rad = 6.366198 turn |
| 50 turn = 314.1593 rad | 50 rad = 7.957747 turn |
| 75 turn = 471.2389 rad | 75 rad = 11.93662 turn |
| 100 turn = 628.3185 rad | 100 rad = 15.91549 turn |
| 150 turn = 942.4778 rad | 150 rad = 23.87324 turn |
| 200 turn = 1,256.637 rad | 200 rad = 31.83099 turn |
| 250 turn = 1,570.796 rad | 250 rad = 39.78874 turn |
| 500 turn = 3,141.593 rad | 500 rad = 79.57747 turn |
| 1000 turn = 6,283.185 rad | 1000 rad = 159.1549 turn |
Turn vs Radian: which is bigger?
A turn is the larger unit: one turn equals 6.283185 radians. Put another way, you need 6.283185 radians to make a single turn.
What is a Turn? History, origin and usage
The turn — also called a revolution, cycle or full circle — is simply one complete rotation, equal to 360 degrees or 2π radians. It is the most intuitive angle “unit,” and the German standard DIN 1315 even proposed a symbol for it; the astronomer Fred Hoyle promoted decimal subdivisions (centiturns, milliturns) in 1962.
The turn is the natural unit for rotational frequency — revolutions per minute (rpm) for engines, motors and disk drives — and for counting coil windings and complete cycles. It underlies how rotation is commonly counted in everyday machinery.
Because one turn equals 2π radians, some mathematicians advocate writing 2π as the single Greek letter τ (tau) so a fraction of a turn reads directly as that fraction of τ. An engine redline of, say, 7,000 rpm means 7,000 complete turns of the crankshaft every minute.
What is a Radian? History, origin and usage
The radian is the angle subtended at a circle’s centre by an arc equal in length to the radius — a concept developed by Roger Cotes around 1714, who recognised its naturalness, though the word “radian” only appeared in print in 1873 in exam questions set by James Thomson at Queen’s College, Belfast. Leonhard Euler had effectively adopted radian measure as early as 1765.
The radian is the SI unit of angle and the natural choice throughout mathematics, physics and engineering, because angular velocity, rotational motion and wave equations come out cleanly only in radians. Calculators and programming languages expect trigonometric functions in radians by default.
A full turn is 2π radians, so one radian is about 57.2958 degrees. Radians make calculus elegant: the derivative of sine equals cosine only when the angle is measured in radians — in degrees an awkward conversion factor appears.
Convert 1 Turn to other units
It can help to see a turn next to the other units it is commonly compared with. One turn (turn) is equal to:
| 1 Turn equals | Unit |
|---|---|
| 6.283185 rad | Radian |
| 360 ° | Degree |
| 400 gon | Gradian |
| 21,600 ′ | Arcminute |
| 1,296,000 ″ | Arcsecond |
Convert 1 Radian to other units
One radian (rad) is equal to:
| 1 Radian equals | Unit |
|---|---|
| 57.29578 ° | Degree |
| 63.66198 gon | Gradian |
| 0.1591549 turn | Turn |
| 3,437.747 ′ | Arcminute |
| 206,264.8 ″ | Arcsecond |
Tips for converting Turn to Radian
- To go from turn to radian, multiply by 6.283185; to go back, divide by 6.283185 (or multiply by 0.1591549).
- For a quick mental estimate, remember that 1 turn ≈ 6.283185 radian, so 10 turn ≈ 62.83185 radian.
- The converter above keeps full precision; the tables round for readability, so use the tool for exact figures.
- Both fields update live as you type — change either side to convert in that direction.
How to use this Turn to Radian converter
Type a number into the Turn field and the equivalent in Radian appears instantly — there is no “convert” button to press and nothing is sent to a server, so it works offline once the page has loaded and is safe for any data. You can also type into the Radian field to convert the other way (radian to turn). Use the swap control to flip the two units, and copy the result with a single click. Because the calculation runs in your browser with full floating-point precision, the converter is more exact than the rounded figures shown in the reference chart above — use the chart for quick look-ups and the tool for precise work.
Accuracy and method
Every figure on this page — the formula, the worked examples, the conversion chart and the “convert to other units” tables — is computed from a single verified conversion factor (1 turn = 6.283185 rad), never copied from a hand-typed lookup table that could drift. That factor is derived from each unit’s definition in terms of a common base unit, and an automated check re-verifies all of the site’s factors on every update, so the numbers here stay correct over time. This matters for turn-to-radian conversions because a small rounding error in a published table compounds quickly at larger quantities; computing from the exact factor avoids that.
Sources
- BIPM — The International System of Units (SI Brochure) — the defining document for the SI, published by the International Bureau of Weights and Measures. It fixes the base unit that this conversion resolves to, which is why the factor here can be exact rather than approximate.
- NIST — Definitions of the SI base units — the US National Institute of Standards and Technology reference for the SI base units, used here as an independent cross-check on the base-unit definition.
Angle factors are exact, anchored to the radian: a full turn = 2π rad = 360° = 400 gon, 1° = 60 arcminutes, 1 arcminute = 60 arcseconds. So the live tool and every number on this page always agree.
Frequently asked questions
- How many radian are in 1 turn?
- 1 turn = 6.283185 radian (rad). To convert any number of turn to radian, multiply by 6.283185.
- How do I convert turn to radian?
- Multiply the number of turn by 6.283185. For example, 10 turn = 10 × 6.283185 = 62.83185 radian, and 25 turn = 157.0796 radian.
- How many turn are in 1 radian?
- 1 radian = 0.1591549 turn. So to convert radian back to turn, multiply by 0.1591549.
- What is the formula to convert turn to radian?
- radian = turn × 6.283185. The reverse formula is turn = radian × 0.1591549.
- How much is 100 turn in radian?
- 100 turn = 628.3185 radian. (And 1000 turn = 6,283.185 radian.)
- Which is bigger, a turn or a radian?
- A turn is larger: 1 turn = 6.283185 radian.
- Is this turn to radian converter accurate?
- Yes — every value is computed from the exact 6.283185 rad per turn factor (not rounded look-up tables), and the factors are re-verified automatically on every update.
- Can I convert radian to turn too?
- Yes. The converter works both ways — enter a value in either field. 1 radian = 0.1591549 turn.