Degrees to radians

What the units mean

Convert degrees (°) to radians (rad) instantly in your browser. See the formula, a worked example and limits of the result.

Angles can be expressed as degrees, radians or complete turns. A full revolution is 360 degrees, 2π radians or one turn. Radians measure the ratio of arc length to radius, which makes them natural in trigonometry and calculus; degrees are familiar in diagrams, navigation and everyday directions.

This page converts one named unit into another within the same physical or digital quantity. The input stays in the browser. The result is a change of representation: it does not estimate an unmeasured property, correct a label, or supply missing information about the object being measured. Identify the original unit before typing a number, especially when a symbol could have more than one informal meaning.

How to calculate

rad = ° × 0.0174532925199. Relationships containing π are exact, but decimal results are rounded for display.

Use the displayed relationship with the entered value in the source unit. For an ordinary linear conversion, multiply by the ratio of the source unit to the destination unit. The reverse direction divides by that same ratio. Fuel economy is the exception here: it uses a reciprocal, so the number appears in the denominator. The calculator carries the defined constants through the calculation before shortening the displayed decimal.

As a worked example, enter 90 °. The output is 1.57079632679 rad. Put the unit beside both numbers when copying them into a document or spreadsheet. A number without its unit does not reveal the measurement scale, and switching the order of the units changes the operation. The example is illustrative; use the value from your own measurement for a real comparison.

When this conversion is useful

Degrees are convenient for human-facing diagrams; radians are expected by most mathematical sine and cosine functions. For example, a right angle is 90° and π/2 radians. If a trigonometric result seems implausible, check whether the function expects radians while the input was entered in degrees. Conversion must happen before evaluation.

Start with a figure explicitly reported in degrees (°) and read the answer in radians (rad). A smaller or larger output number alone says nothing about whether the measured quantity has changed. If the original source gives a rounded value, the calculation cannot recover digits that were never measured. For a critical limit, compare using the unrounded numerical result and round only the final presentation.

Precision and common mistakes

The conversion is proportional, but factors containing π cannot be written as finite decimals. Calculator output is therefore an approximation even when the geometric relationship is exact. Negative angles describe the opposite rotation direction under a chosen sign convention. More than one complete turn can be useful when counting rotation rather than final orientation.

Conversion factors can be defined exactly even when the decimal displayed on the page has been shortened. Values involving π are approximated by computer arithmetic; a stated measurement may have much greater uncertainty than that arithmetic. Carry the source unit and enough significant figures through intermediate steps. If you need a certified measurement, calibration or commercial settlement, use the applicable standard and the original instrument record.

Questions about the result

Why can a reverse conversion return a slightly different final digit? The visible result is rounded to make it readable; reversing that shortened number starts from a different input. Why does a plausible figure sometimes disagree with a product label? The label may use another convention, a different reference condition or fewer significant figures. Confirm the exact unit symbol and the measurement definition before deciding that either number is wrong.