Radians to Degrees Converter
Free Radians degrees Calculator for angles. Enter values to get step-by-step solutions with formulas and graphs. Get results you can export or share.
Calculator
Adjust values & calculateCommon Angle Reference
Formula
To convert radians to degrees, multiply by 180/pi (approximately 57.2958). To convert degrees to radians, multiply by pi/180 (approximately 0.01745). This relationship comes from the fact that a full circle is both 360 degrees and 2*pi radians.
Last reviewed: December 2025
Worked Examples
Example 1: Converting Pi/4 Radians to Degrees
Example 2: Converting 2.5 Radians to Degrees
Background & Theory
The Radians to Degrees Converter applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Radians to Degrees Converter traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Sources & References
Formula
Degrees = Radians x (180 / pi)
To convert radians to degrees, multiply by 180/pi (approximately 57.2958). To convert degrees to radians, multiply by pi/180 (approximately 0.01745). This relationship comes from the fact that a full circle is both 360 degrees and 2*pi radians.
Worked Examples
Example 1: Converting Pi/4 Radians to Degrees
Problem: Convert pi/4 radians to degrees, DMS notation, and find the trig values.
Solution: Degrees = (pi/4) x (180/pi) = 45 degrees\nDMS = 45 deg 0' 0\"\nsin(pi/4) = sqrt(2)/2 = 0.7071\ncos(pi/4) = sqrt(2)/2 = 0.7071\ntan(pi/4) = 1.0000\nQuadrant: I
Result: pi/4 radians = 45 degrees (Quadrant I)
Example 2: Converting 2.5 Radians to Degrees
Problem: Convert 2.5 radians to degrees and identify the quadrant.
Solution: Degrees = 2.5 x (180/pi) = 2.5 x 57.2958 = 143.239 degrees\nDMS = 143 deg 14' 20.14\"\nNormalized: 143.239 degrees\nQuadrant: II (between 90 and 180)\nsin(2.5) = 0.5985, cos(2.5) = -0.8011
Result: 2.5 radians = 143.239 degrees (Quadrant II)
Frequently Asked Questions
What is the relationship between radians and degrees?
Radians and degrees are two different units for measuring angles. One complete revolution around a circle equals 360 degrees or 2*pi radians (approximately 6.2832 radians). Therefore, 1 radian equals 180/pi degrees (approximately 57.2958 degrees), and 1 degree equals pi/180 radians (approximately 0.01745 radians). The conversion formulas are: degrees = radians times (180/pi) and radians = degrees times (pi/180). Radians are dimensionless because they represent the ratio of arc length to radius, making them the natural unit for angle measurement in mathematics and physics.
Why do mathematicians prefer radians over degrees?
Mathematicians prefer radians because they simplify many formulas and make calculus cleaner. When angles are in radians, the derivative of sin(x) is simply cos(x), and the derivative of cos(x) is -sin(x). In degrees, these derivatives require an extra factor of pi/180. The Taylor series for sin(x) = x - x^3/3! + x^5/5! - ... only works when x is in radians. Arc length on a circle is simply s = r*theta when theta is in radians. The small angle approximations sin(x) approximately equals x and tan(x) approximately equals x only hold in radians. Euler's formula e^(ix) = cos(x) + i*sin(x) requires radians. Essentially, radians eliminate conversion factors from nearly every trigonometric and calculus formula.
How do you convert DMS (degrees, minutes, seconds) to radians?
To convert DMS notation to radians, first convert everything to decimal degrees, then multiply by pi/180. For example, convert 45 degrees 30 minutes 15 seconds: the minutes component equals 30/60 = 0.5 degrees, the seconds component equals 15/3600 = 0.004167 degrees. Total decimal degrees = 45 + 0.5 + 0.004167 = 45.504167 degrees. Multiply by pi/180: 45.504167 times 0.017453 = 0.79418 radians. Going the reverse direction: convert radians to decimal degrees (multiply by 180/pi), then extract degrees as the integer part, multiply the fractional part by 60 for minutes, and multiply the fractional minutes by 60 for seconds.
How are radians used in physics and engineering?
Radians are the standard angle unit in physics and engineering equations. Angular velocity is measured in radians per second (rad/s), not degrees per second. The relationship v = r*omega (linear velocity equals radius times angular velocity) requires omega in radians. Torque equations, moment of inertia calculations, and rotational dynamics all use radians. In electrical engineering, AC voltage is described by V(t) = V0*sin(omega*t), where omega (angular frequency) is in radians per second. Phase differences between signals are expressed in radians. Structural engineers use radians when calculating beam deflections and structural analysis involving trigonometric functions.
What are gradians and how do they compare to radians and degrees?
Gradians (also called gons or grads) are a third angle measurement system where a right angle equals 100 gradians and a full circle equals 400 gradians. One gradian equals 0.9 degrees or pi/200 radians. Gradians were introduced during the French Revolution as part of the metric system and are still used in some European countries for surveying and civil engineering. The conversion formulas are: gradians = degrees times (10/9) and gradians = radians times (200/pi). While gradians make right angles and percentages of a turn easy to express (a right angle is exactly 100 grad), they never gained the mathematical elegance of radians or the widespread familiarity of degrees.
Can radians be negative or greater than 2pi?
Yes, radian measures can be any real number, positive, negative, or beyond 2pi. Positive angles are measured counterclockwise from the positive x-axis, while negative angles go clockwise. So -pi/2 radians is 90 degrees clockwise, equivalent to 270 degrees or 3pi/2 radians counterclockwise. Angles greater than 2pi represent multiple revolutions: 3pi radians is 1.5 revolutions (540 degrees), and 4pi radians is 2 complete revolutions (720 degrees). Coterminal angles differ by multiples of 2pi and share the same trigonometric values. In physics, cumulative angles matter: a wheel rotating 10pi radians has made 5 complete turns, even though its angular position is equivalent to 0 radians.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy