Degrees to Radians Converter
Our free angles calculator solves degrees radians problems. Get worked examples, visual aids, and downloadable results.
Calculator
Adjust values & calculateCommon Angles Reference
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | pi/6 | 1/2 | sqrt(3)/2 | 1/sqrt(3) |
| 45° | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60° | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90° | pi/2 | 1 | 0 | undefined |
| 120° | 2pi/3 | sqrt(3)/2 | -1/2 | -sqrt(3) |
| 135° | 3pi/4 | sqrt(2)/2 | -sqrt(2)/2 | -1 |
| 150° | 5pi/6 | 1/2 | -sqrt(3)/2 | -1/sqrt(3) |
| 180° | pi | 0 | -1 | 0 |
| 270° | 3pi/2 | -1 | 0 | undefined |
| 360° | 2pi | 0 | 1 | 0 |
Formula
Since a full circle is both 360 degrees and 2*pi radians, the conversion factor is pi/180 (about 0.01745) for degrees to radians, and 180/pi (about 57.296) for radians to degrees. One radian equals the angle where the arc length equals the radius.
Last reviewed: December 2025
Worked Examples
Example 1: Converting 135 Degrees to Radians
Example 2: Converting 2.5 Radians to Degrees
Background & Theory
The Degrees to Radians 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 Degrees to Radians 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
Formula
radians = degrees * (pi/180), degrees = radians * (180/pi)
Since a full circle is both 360 degrees and 2*pi radians, the conversion factor is pi/180 (about 0.01745) for degrees to radians, and 180/pi (about 57.296) for radians to degrees. One radian equals the angle where the arc length equals the radius.
Worked Examples
Example 1: Converting 135 Degrees to Radians
Problem: Convert 135 degrees to radians and express as a fraction of pi.
Solution: 135 * (pi/180) = 135*pi/180\nSimplify: GCD(135, 180) = 45\n135/45 = 3, 180/45 = 4\nResult: 3*pi/4 = 3 * 3.14159 / 4 = 2.3562 radians\nQuadrant: II (between 90 and 180)\nReference angle: 180 - 135 = 45 degrees = pi/4
Result: 135 degrees = 3*pi/4 = 2.3562 radians (Quadrant II, reference angle 45 degrees)
Example 2: Converting 2.5 Radians to Degrees
Problem: Convert 2.5 radians to degrees and find the quadrant.
Solution: 2.5 * (180/pi) = 2.5 * 57.2958 = 143.239 degrees\nQuadrant: II (between 90 and 180)\nReference angle: 180 - 143.239 = 36.761 degrees\nsin(2.5) = 0.5985, cos(2.5) = -0.8011\nBoth signs confirm Quadrant II (sin positive, cos negative).
Result: 2.5 radians = 143.239 degrees (Quadrant II). sin = 0.599, cos = -0.801.
Frequently Asked Questions
How do you convert degrees to radians?
To convert degrees to radians, multiply the degree value by pi/180. This conversion factor comes from the fact that a full circle is both 360 degrees and 2*pi radians, so the ratio is 2*pi/360 = pi/180. For example, 90 degrees = 90 * (pi/180) = pi/2 radians. Another way to think about it: divide the degree measure by 180, then multiply by pi. Common conversions worth memorizing include 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, and 180 degrees = pi radians. These relationships form the foundation for working with trigonometric functions in mathematics and science.
Why do mathematicians prefer radians over degrees?
Radians are preferred in mathematics because they create the simplest possible formulas in calculus and analysis. The derivative of sin(x) equals cos(x) only when x is in radians. If x were in degrees, the derivative would include an extra factor of pi/180. Similarly, the fundamental limit lim(sin(x)/x) = 1 as x approaches 0 only holds in radians. The Taylor series for trigonometric functions also require radians: sin(x) = x - x^3/6 + x^5/120 - ... only works for radians. Arc length formula s = r*theta is cleanest in radians with no conversion factor. The Euler formula e^(i*theta) = cos(theta) + i*sin(theta) also requires radians. These mathematical simplifications make radians the natural choice for analysis.
What are gradians and how do they relate to degrees and radians?
Gradians (also called grads or gons) divide a right angle into 100 parts, making a full circle 400 gradians. This metric-inspired system was introduced during the French Revolution as part of the metrification effort. Conversions: 1 degree = 10/9 gradians, 1 gradian = 0.9 degrees = pi/200 radians. Gradians are primarily used in surveying and some European engineering applications because 100 gradians equals a right angle, making percentage-of-slope calculations easier. However, gradians never achieved widespread adoption in mathematics or physics, where radians dominate. Scientific calculators typically offer three angle modes: degrees (DEG), radians (RAD), and gradians (GRAD), so always verify which mode is active before computing.
What is a turn and how does it compare to degrees and radians?
A turn (also called a revolution or cycle) is the simplest angular unit: one turn equals a full rotation. Half a turn is a semicircle, quarter turn is a right angle. Conversions: 1 turn = 360 degrees = 2*pi radians = 400 gradians. Turns are intuitive for describing rotational motion: a wheel making 3.5 turns is easier to visualize than 1260 degrees or 7*pi radians. In some programming languages and graphics frameworks (notably LOGO), turns are the primary angle unit. The tau movement (tau = 2*pi = 6.2832) advocates using tau instead of pi because 1 tau equals exactly 1 turn, making many formulas more intuitive. For example, a quarter circle is tau/4 rather than pi/2.
Can I use Degrees to Radians Converter on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
How do I verify Degrees to Radians Converter's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy