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To convert from degree (deg) to radian (rad), use the following formula:
radian (rad)
= π180× degree (deg)
= 0.017453292519943294444× degree (deg)
Let's convert 5 degree (deg) to radian (rad).
Using the formula:
5 × 0.017453292519943294444 = 0.08726646259971647222
Therefore, 5 degree (deg) is equal to 0.08726646259971647222 radian (rad).
How many degrees are in one radian? One radian (rad) contains 57.295779513082325226 degrees (deg) — the inverse of the factor above. Multiplying by 0.017453292519943294444 takes you from degrees to radians; multiplying by 57.295779513082325226 brings you back.
Put in words: one degree equals 0.017453292519943294444 radians, so the degree is the smaller unit of this pair. Converting between them never changes the amount of angle being measured — only the size of the unit you count it in.
In geometry and everyday life, we measure angles using degrees.
A degree (represented by the universal symbol °) is the basic unit for measuring rotation. It helps us describe the amount of turn between two lines that meet at a point.
For instance, a perfect corner, like the edge of a book, is a 90° right angle, and one complete, full-circle rotation measures 360°.
The standard of a circle containing 360 degrees (360°) originated with the ancient Babylonians.
They used a base-60 (sexagesimal) number system, and 360 was a perfect number for them because it is highly divisible. Its large number of factors made it incredibly easy to perform fractional calculations in fields like astronomy and geometry.
For fields that require extreme precision, like astronomy, cartography (map-making), and GPS navigation, a single degree is broken down into even smaller units:
This system ensures that even the tiniest angles can be measured with high accuracy.
There is no physical reason a full turn must be 360 of anything. The number is an inheritance from Babylonian astronomy, and it survived for two reasons that still matter.
The first is the calendar. Babylonian astronomers worked with a year of roughly 360 days, so the Sun appeared to advance about one degree along the ecliptic each day — a convenient bookkeeping unit for tracking the sky.
The second is arithmetic, and it is the reason the convention outlived the astronomy. 360 has 24 divisors. It splits evenly into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths, twelfths, fifteenths, eighteenths, twentieths, twenty-fourths and more. Every common fraction of a turn lands on a whole number of degrees: a right angle is 90°, a third of a turn is 120°, an eighth is 45°. A base-10 alternative such as the gradian, with 400 to a circle, is better for decimal arithmetic but produces fractions for thirds and sixths — which is roughly why it never displaced the degree outside surveying.
Degrees are for people; radians are for mathematics and for machines.
The distinction is not stylistic. In calculus the derivative of sin(x) is cos(x) only when x is in radians; in degrees an awkward factor of π/180 appears and never leaves. Because of this, essentially every programming language's trigonometric library — JavaScript, Python, C, Java — takes radians, not degrees.
That mismatch is a reliable source of bugs. A value that looks plausible but is wrong by a factor of about 57.3 usually means degrees were passed where radians were expected. The conversions worth memorising are:
Scientific calculators expose the same trap through their DEG / RAD / GRAD mode switch, where an unnoticed mode is the classic cause of an answer that is confidently and completely wrong.
A radian (rad) is the standard unit used to measure angles, and it's part of the International System of Units (SI).
One radian is equal to roughly 57.3 degrees (deg), or more precisely, 180/π degrees.
The radian is the official SI unit for angles, making it the preferred choice in science and engineering.
It is especially common in fields like physics, calculus, and computer graphics. The main advantages of using radians are:
A full circle has 360 degrees (360°), which is equal to 2π radians.
This key relationship gives us the precise formulas for converting between radians and degrees:
Degrees = Radians × (180/π)Radians = Degrees × (π/180)Here are some common angle conversions:
| Degrees | Radians (exact) | Radians (approx.) |
|---|---|---|
| 30° | π/6 rad | 0.524 rad |
| 45° | π/4 rad | 0.785 rad |
| 60° | π/3 rad | 1.047 rad |
| 90° | π/2 rad | 1.571 rad |
| 180° | π rad | 3.142 rad |
| 360° | 2π rad | 6.283 rad |
Geometrically, the definition of a radian is based on a circle's own properties.
Imagine an arc along the edge of a circle that has the same length as the circle's radius. The angle created at the center of the circle by this specific arc is exactly one radian.
Because a radian is a ratio of two lengths (arc length divided by radius), it is technically a dimensionless unit. This unique property is why it integrates so smoothly into advanced math and physics formulas.
Here are some quick reference conversions from degree (deg) to radian (rad):
| degrees | radians |
|---|---|
| 0.000001 deg | 1.7453292519943294444× 10-8 rad |
| 0.001 deg | 0.000017453292519943294444 rad |
| 0.1 deg | 0.0017453292519943294444 rad |
| 1 deg | 0.017453292519943294444 rad |
| 2 deg | 0.034906585039886588888 rad |
| 3 deg | 0.052359877559829883332 rad |
| 4 deg | 0.069813170079773177776 rad |
| 5 deg | 0.08726646259971647222 rad |
| 6 deg | 0.10471975511965976666 rad |
| 7 deg | 0.12217304763960306111 rad |
| 8 deg | 0.13962634015954635555 rad |
| 9 deg | 0.15707963267948965 rad |
| 10 deg | 0.17453292519943294444 rad |
| 20 deg | 0.34906585039886588888 rad |
| 30 deg | 0.52359877559829883332 rad |
| 40 deg | 0.69813170079773177776 rad |
| 50 deg | 0.8726646259971647222 rad |
| 100 deg | 1.7453292519943294444 rad |
| 1000 deg | 17.453292519943294444 rad |
| 10000 deg | 174.53292519943294444 rad |