Degrees to Radians Converter
Our free angles calculator solves degrees radians problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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 Simplify: GCD(135, 180) = 45 135/45 = 3, 180/45 = 4 Result: 3*pi/4 = 3 * 3.14159 / 4 = 2.3562 radians Quadrant: II (between 90 and 180) Reference 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 Quadrant: II (between 90 and 180) Reference angle: 180 - 143.239 = 36.761 degrees sin(2.5) = 0.5985, cos(2.5) = -0.8011 Both 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 is a radian and where does the definition come from?
A radian is the angle subtended at the center of a circle by an arc whose length equals the radius of the circle. Since the circumference of a circle is 2*pi*r, a full circle contains 2*pi radians (approximately 6.2832 radians). One radian is approximately 57.296 degrees. The concept was first described by Roger Cotes in 1714, though the term 'radian' was not coined until 1873 by James Thomson. Radians are dimensionless (arc length divided by radius gives a pure number), which makes them particularly useful in physics where they can be dropped from or added to equations without affecting dimensional analysis. This dimensionless nature is both a strength and a potential source of confusion.
What are the radian values of common angles?
The most important angle conversions to memorize are: 0 degrees = 0 radians, 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, 120 degrees = 2*pi/3, 135 degrees = 3*pi/4, 150 degrees = 5*pi/6, 180 degrees = pi, 210 degrees = 7*pi/6, 225 degrees = 5*pi/4, 240 degrees = 4*pi/3, 270 degrees = 3*pi/2, 300 degrees = 5*pi/3, 315 degrees = 7*pi/4, 330 degrees = 11*pi/6, and 360 degrees = 2*pi. These angles appear repeatedly on the unit circle and in trigonometry courses. Noticing the pattern that the denominator relates to the fraction of 180 degrees helps with quick mental conversions.
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.
How does angle measurement relate to the unit circle?
The unit circle (a circle with radius 1 centered at the origin) is the geometric foundation for understanding both degrees and radians. An angle in standard position starts from the positive x-axis and opens counterclockwise. On the unit circle, the radian measure of an angle equals the length of the arc from (1,0) to the terminal point. Since the unit circle has circumference 2*pi, a full revolution has arc length 2*pi, explaining why 360 degrees equals 2*pi radians. The coordinates of any point on the unit circle are (cos(theta), sin(theta)), directly connecting angle measurement to trigonometric functions. This is why the unit circle is the primary tool for understanding trigonometry.
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.
What common mistakes do students make with degree-radian conversions?
The most frequent error is forgetting to set the calculator to the correct mode (degrees or radians) before computing trigonometric functions. Computing sin(90) in radian mode gives 0.8940 instead of the expected 1, because the calculator interprets 90 as radians rather than degrees. Another common mistake is multiplying by pi/180 when converting radians to degrees instead of by 180/pi. Students also confuse the formulas: remember that converting TO a smaller unit (radians are about 57 times larger than degrees) requires multiplying by a fraction less than 1, specifically pi/180. Finally, when simplifying radian expressions, students sometimes incorrectly cancel pi or forget that pi is approximately 3.14159, not exactly 3.
How are degree-radian conversions used in programming?
Most programming languages compute trigonometric functions in radians, requiring explicit conversion from degrees. In JavaScript, Math.sin(angle * Math.PI / 180) converts degrees to radians before computing sine. Python similarly requires math.radians(degrees) or manual multiplication. CSS is an exception, supporting both deg and rad units directly: transform: rotate(45deg) or rotate(0.785rad). Game engines like Unity use degrees in their inspector but radians internally, with built-in conversion functions like Mathf.Deg2Rad. When interfacing with GPS coordinates (in degrees) and computing distances using the Haversine formula (which requires radians), proper conversion is essential. A wrong conversion mode in navigation software once contributed to actual flight incidents.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Radians to Degrees Converter
Calculate radians to degrees converter with inputs, formulas, and instant results.
🧮Polish Notation Converter (Prefix, Infix, Postfix)
Calculate polish notation converter with inputs, formulas, and instant results.
🧮Binary Fraction Converter
Calculate binary fraction converter with inputs, formulas, and instant results.
🧮Gray Code Converter
Calculate gray code converter with inputs, formulas, and instant results.
🧮IEEE 754 Floating-Point Converter
Calculate ieee754floating point converter with inputs, formulas, and instant results.
🧮Angle Converter
Calculate angle converter with inputs, formulas, and instant results.
🧮Data Storage Converter
Calculate data storage converter with inputs, formulas, and instant results.
🧮Energy Converter
Calculate energy converter with inputs, formulas, and instant results.