Convert radians to degrees from a decimal or a simple pi expression, then check the exact form, decimal form, turns, and matching angle.
Advanced options
Table of contents
How to use our Radians to Degrees Converter
- Enter Angle in radians as a decimal like 1.2 or a simple pi expression like pi/6, 3pi/4, -2pi, or 1.5pi.
- Choose Show degrees as to make the main answer favor an exact degree value when the input is simple, or a rounded decimal answer.
- Set Decimal places if you want a different rounding level for Decimal degree form. Leaving it blank uses 4.
- Click Calculate to see Angle in degrees, plus Exact degree form, Decimal degree form, Same angle in full turns, and Matching angle from 0 to less than 360 degrees when useful.
- Sanity-check the result: pi should equal 180 degrees, pi/2 should equal 90 degrees, and a negative input should keep a negative Angle in degrees even though the matching 0 to less than 360 degree angle may be positive.
Definitions
Radian (rad): A unit for measuring angles based on a circle's radius. Radians and degrees are two common angle units [2].
Degree (deg): Another angle unit. A full circle is 360 degrees.
Pi: The circle constant written as pi, about 3.141592653589793. In angle work, pi radians equals 180 degrees.
Exact degree form: A degree answer written without rounding when the radian input is a simple multiple of pi, such as 30 degrees from pi/6.
Decimal degree form: The degree answer rounded to the number of Decimal places you choose.
Full turns: The same angle measured as part of a full circle, where 1 turn equals 360 degrees.
Coterminal angle: An angle that points in the same direction after adding or subtracting full turns. This calculator shows one coterminal angle from 0 to less than 360 degrees.
Common mistakes and quick fixes
Mistake: Leaving Angle in radians blank.
Fix: Enter a value such as 0, 1.2, pi/6, 3pi/4, or -2pi before you calculate.
Mistake: Typing an invalid expression in Angle in radians , such as pi//2, 2**pi, or extra letters.
Fix: Use only numbers, pi, one slash for a fraction, a leading minus sign if needed, and simple forms like pi/3 or 1.5pi.
Mistake: Expecting Exact degree form for a decimal-only input like 1.234.
Fix: Use a simple pi-based input if you want an exact answer, or read Decimal degree form for decimal inputs.
Mistake: Thinking Matching angle from 0 to less than 360 degrees is the main converted answer.
Fix: Read Angle in degrees first. The matching angle is just the wrapped version that points the same direction.
Mistake: Entering a negative or very large angle and assuming the result must stay between 0 and 360.
Fix: Keep Angle in degrees as the true converted value. Use Matching angle from 0 to less than 360 degrees only as a helper.
Mistake: Entering a bad value for Decimal places , such as text or a negative number.
Fix: Leave Decimal places blank for 4, or enter a whole number from 0 to 30.
Limitations & Key Assumptions / Boundary Conditions
- Angle in radians accepts decimals and simple pi expressions only. It does not handle full algebra, functions, powers, or complex nested expressions.
- Exact degree form appears only when the input can be recognized as a simple multiple of pi. Decimal-only inputs usually show only decimal degrees.
- Decimal degree form depends on the chosen Decimal places, so rounded values can differ slightly from textbook or app answers that use a different rounding rule.
- Angle in degrees is not automatically wrapped into one turn. Large positive or negative inputs stay large or negative by design.
- Matching angle from 0 to less than 360 degrees is a helper for direction checking, not a replacement for the main converted angle.
- Decimal inputs and decimal coefficients in simple pi expressions are parsed exactly for displayed precision. Plain decimal-radian conversions use 100 decimal digits of pi as lower and upper bounds, and a result is shown only when both bounds round to the same requested value.
- Inputs so large or so close to a rounding or full-turn boundary that the requested digits cannot be certified are rejected with guidance instead of displaying uncertain digits.
Methodology
Conversion rule
The calculator converts radians to degrees with the standard relationship that pi radians equals 180 degrees [1][2].
degrees = radians * (180 / pi)
Supporting outputs
After finding degrees, it also shows the angle as turns and gives one coterminal angle from 0 to less than 360 degrees.
turns = degrees / 360
coterminal = ((degrees % 360) + 360) % 360
How pi expressions are handled
If Angle in radians is a simple pi expression such as pi/6, 3pi/4, -2pi, or 1.5pi, the calculator first reads it as a multiple of pi. That makes it possible to show an Exact degree form when the result is simple. If the input is a decimal like 1.2, the calculator evaluates it numerically and shows a decimal degree answer.
Mini example
For pi/6 radians:
degrees = (pi / 6) * (180 / pi) = 30
turns = 30 / 360 = 0.083333...
coterminal = 30
For -3pi/4 radians:
degrees = (-3pi / 4) * (180 / pi) = -135
coterminal = 225
Parsing and rounding choices
Decimal places controls only the rounded decimal display. Leaving it blank uses 4. Non-blank values are expected to be whole numbers, and the calculator limits them to a safe range of 0 to 30. The main answer stays the true converted angle; it is not automatically normalized into one turn.
Assumptions and limits
The parser is intentionally narrow so it can reject unclear input. It allows decimals and simple pi-based forms, but it rejects unsupported characters, ambiguous comma grouping, repeated slashes, divide-by-zero cases, and exponent operators. Exact simplification is attempted only for simple recognizable pi multiples, so some valid numeric inputs will show decimal output only. For decimal output, the calculator keeps entered decimals as exact BigInt fractions. Decimal-radian conversions enclose pi between consecutive 100-decimal bounds and accept a rounded result only when both bounds agree, providing 70 guard decimal places for a 30-place display under normal input sizes.