Degrees to Radians Converter — Interactive Unit Circle with Exact π Fractions

What is a radian?

A radian is the natural unit of angular measure. Take a circle of any size and consider an arc along its circumference whose length is exactly equal to the circle's radius. The angle that this arc subtends at the centre of the circle is one radian. That is the whole definition — no arbitrary constant, no historical accident, just the geometry of a circle measuring itself.

Because the circumference of a circle is 2π × radius, a full trip around the circle is 2π radians. This gives the fundamental equivalence that anchors every degree-to-radian conversion: 2π radians = 360°, or dividing both sides by 2, π radians = 180°. Every other common conversion is derived from that one fact.

Radians are the standard unit of angular measure throughout mathematics, physics and engineering. They make calculus formulas cleaner — the derivative of sin is cos only when the angle is in radians — they make arc length trivial (s = rθ with no extra factor), and they make rotation equations in physics dimensionally consistent. Degrees remain useful for everyday intuition and for navigation and surveying, where the Babylonians' base-60 division of the circle has survived for over three thousand years.

The conversion formula — and why it works

Degrees-to-radians conversion uses one factor. From π radians = 180°, divide both sides by 180:

So to convert any angle:

The same factor π/180 appears everywhere degrees and radians meet. Every other unit conversion on this page — gradians, turns, arcminutes, arcseconds, milliradians — is built from the same principle with a different denominator.

Understanding the unit circle

The unit circle is a circle of radius 1 centred on the origin of the Cartesian plane. Every angle in trigonometry corresponds to one point on this circle. The x-coordinate of that point is the cosine of the angle, and the y-coordinate is the sine. The tangent is the slope of the ray from the origin to the point.

The interactive visual in this tool shows a right triangle inscribed in the unit circle. The horizontal leg (in blue) has length equal to cos θ, the vertical leg (in pink) has length equal to sin θ, and the hypotenuse (in indigo) has length 1 — because the radius of the unit circle is 1. This is the geometric meaning of sine and cosine: they are the legs of a right triangle whose hypotenuse is the radius.

The four quadrants

Walking anticlockwise from the positive x-axis covers the four quadrants in order. Each quadrant has a characteristic set of signs for the six trig functions:

The mnemonic ASTC — All Students Take Calculus — encodes which functions are positive in each quadrant, going anticlockwise from the first.

Angles wrap around the circle. An angle of 405° is the same physical direction as 45° because 405 = 360 + 45. An angle of −30° is the same direction as 330° because −30 + 360 = 330. The interactive circle wraps any input to the range 0°–360° so you always see the actual direction.

Reference angles

A reference angle is the acute angle (0°–90°) that the terminal side of an angle makes with the x-axis. For example:

• 150° — the terminal side is 30° from the negative x-axis, so the reference angle is 30°.
• 225° — the terminal side is 45° from the negative x-axis, so the reference angle is 45°.
• 300° — the terminal side is 60° from the positive x-axis, so the reference angle is 60°.

Reference angles let you find trig values for any angle from the values at the first-quadrant angle. Combined with the ASTC sign rule, this is how you evaluate sin(150°), cos(225°), tan(300°) and any other angle in any quadrant.

The eight angles you should memorise

These eight angles cover almost every problem in trigonometry, calculus and physics. Learn the degree value and the matching π fraction, and you can build the rest by halving or doubling.

💡 Memory trick. Starting from 0° and going up to 90°, the sine values are √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, ½, √2/2, √3/2, 1. The cosine is the same list in reverse. Once you know that pattern you can derive every special-angle trig value in your head.

Worked examples

Example 1 — 90° to radians

Example 2 — 45° to radians

Example 3 — 270° to radians

Example 4 — π/6 radians to degrees

Example 5 — 1 radian to degrees

Example 6 — 2.5 radians to degrees

Example 7 — Negative angle: −60° to radians

Example 8 — Large angle: 720° to radians

Example 9 — 15° to radians (rare, uses a compound surd for tan)

Example 10 — 22.5° to radians (half of 45°)

Why mathematicians and physicists prefer radians

Radians are not just another way of writing angles. They are the unit that makes calculus, physics and Fourier analysis work cleanly. A few concrete examples show why:

• Derivative of sine. The derivative of sin(x) is cos(x), but only when x is measured in radians. If x is in degrees, the derivative picks up a factor of π/180 — messy and easy to forget. The Taylor series sin(x) = x − x³/6 + x⁵/120 − … is only correct for x in radians.
• Arc length. Arc length along a circle of radius r, subtended by angle θ, is s = rθ — with no additional constant — when θ is in radians.
• Angular velocity. In physics, angular velocity ω measured in radians per second connects directly to linear velocity via v = rω.
• Small-angle approximations. sin(θ) ≈ θ, tan(θ) ≈ θ and cos(θ) ≈ 1 − θ²/2 are only valid when θ is in radians. This is why pendulum formulas and optics equations use radians without saying so.
• Rotational kinetic energy and moment of inertia. Everything from angular momentum (L = Iω) to rotational kinetic energy (E = ½Iω²) assumes ω is in radians per second.
• Fourier transforms and complex exponentials. The formula e^iθ = cos(θ) + i·sin(θ) is only correct for θ in radians.

None of this means degrees are wrong. Degrees are perfectly useful for everyday geometry, navigation, cartography and artillery. They divide the circle into a number (360) that has an unusually large number of divisors — 24 of them — which makes them extremely convenient for hand calculation and reading compasses. Both units are correct; they serve different purposes.

Other angle units

Gradians (gon)

A gradian divides a right angle into 100 parts, so a full circle is 400 gradians. The unit was introduced during the French Revolution as part of a broader attempt to decimalise measurement. Gradians survive in European surveying and in some military artillery calculations because the 100-gradian right angle makes mental arithmetic easier. 1 gradian = 0.9°, so 100 gradians = 90°.

Turns (revolutions)

A turn, revolution, or full rotation is exactly one circle: 360° or 2π radians. Turns are natural for rotation counts, rotation frequencies (a motor spinning at 3,000 rpm completes 50 turns per second) and periodic signals. Many programming environments and graphics libraries use turns as a first-class angle unit because the number has no arbitrary offset — a full circle is simply 1.

Arcminutes and arcseconds

Arcminutes and arcseconds are sub-divisions of a degree, using the same base-60 principle that gives us hours and minutes of time. One degree contains 60 arcminutes, and one arcminute contains 60 arcseconds. So one degree equals 3,600 arcseconds. Astronomers use these units to record small angular distances between stars and planets; a typical telescope field of view is measured in arcminutes, and the apparent diameter of a planet may be measured in arcseconds. Navigation and surveying use the same units — a position is often written as 51° 30′ 26″ North.

Milliradians (mrad)

A milliradian is one-thousandth of a radian, and it is the favourite unit of artillery and precision optics because it makes small-angle estimation trivial. Near the small-angle limit, an object that subtends 1 milliradian at a distance of 1,000 metres has a lateral size of about 1 metre. This "1 mrad ≈ 1 metre at 1 kilometre" rule lets a range estimator or a shooter work out distances from known object sizes without a calculator. 1 radian = 1,000 mrad, so 1 degree ≈ 17.4533 mrad.

Where degree-to-radian conversion matters

• Trigonometry and pre-calculus coursework. Every problem involving special angles requires knowing the radian equivalents of 30°, 45° and 60°.
• Calculus. Limits, derivatives of trigonometric functions and Taylor series all require radian measure to work with standard formulas.
• Physics. Circular motion, harmonic oscillation, wave equations and rotational dynamics all use radians as the base unit.
• Programming and graphics. Most programming languages' trigonometric functions expect radians; screen coordinates and rotation matrices use radians for camera control, sprite rotation and 3D transforms.
• Navigation and GIS. Geographic Information Systems and GPS use radians for spherical trigonometry.
• Astronomy and celestial mechanics. Orbital positions, right ascension and declination are frequently expressed in radians for computation.
• Signal processing and DSP. Fourier transforms, filter design and phase calculations all rely on radian measure.
• Robotics. Joint angles in robotics libraries like ROS and MoveIt use radians as the canonical unit.
• Computer games and 3D engines. Unity, Unreal, Godot and most engines use radians for their internal math.
• Astronomy instrument control. Telescope mounts, star trackers and planetarium software all use radians for the underlying calculations.

Converting in Excel, Google Sheets, Python and JavaScript

💡 Language trap. Python's math module has radians and degrees but no deg2rad. MATLAB and Octave use deg2rad and rad2deg. JavaScript has neither and requires the raw arithmetic. If you are moving code between languages, always check the function name rather than assuming.

Common mistakes when converting degrees to radians

• Using the wrong conversion factor. Multiplying by π/180 shrinks the number (π/180 ≈ 0.01745); multiplying by 180/π enlarges it (180/π ≈ 57.3). Radians are numerically smaller than degrees for the same angle.
• Calculator mode mismatch. Most scientific calculators have DEG, RAD and sometimes GRAD modes. If the calculator is in RAD mode and you type sin(30), it computes sin(30 radians), not sin(30°).
• Forgetting that π is not 3.14. Using 3.14 instead of the full-precision value introduces small errors that compound.
• Assuming all angles reduce to a nice π fraction. Only angles that are multiples of 15° reduce to a fraction of π with a small denominator.
• Mixing radians and degrees inside one expression. A formula like sin(90) + cos(π/2) mixes degrees and radians — the results will not cancel as you might expect.
• Negative angle confusion. Negative angles represent clockwise rotation. The conversion factor is applied identically. −45° × π/180 = −π/4.
• Forgetting to wrap after multiple revolutions. 810° is physically the same as 90° because 810 = 720 + 90.
• Reading radians as if they were degrees. A result like "0.7854" is easy to misread as 0.7854°. It is actually π/4 ≈ 0.7854 radians, which is 45°.

A brief history of angular measurement

The degree is a very old unit — its 360-part division of the circle appears in Babylonian astronomy around 2,000 BCE. Historians disagree on the exact origin. The most common theory is that 360 was chosen because it approximates the number of days in a year (the sun moves roughly one degree per day along the ecliptic) and because 360 has an unusually large number of divisors (24 in total). The division of each degree into 60 minutes and each minute into 60 seconds came later, from the same base-60 arithmetic that gave us hours and minutes of time.

The radian is far more recent. The concept appears implicitly in the work of Roger Cotes in 1714, and the term was first used in print by James Thomson in 1873. The unit became standard in mathematics during the 19th century as the calculus of trigonometric functions was formalised — the derivative of sine being cosine only in radian measure made radians the natural unit for analysis.

The gradian is a much smaller chapter. It was introduced during the French Revolution as part of a systematic attempt to decimalise all measurement. It survived in French surveying and briefly in French military artillery, and remnants of it are still visible in some European surveying equipment.

Privacy and data storage

Every calculation on this page runs entirely in your browser using vanilla JavaScript. Nothing you type is uploaded, logged, cached or transmitted anywhere. There is no account, no login and no network request after the page loads. The recent-angles history is stored only in your browser's local storage and can be cleared with a single click.

Limitations

• The exact π fraction display covers denominators of 12 or fewer. Angles like 7° (which is π × 7/180) show only the decimal because the fraction is not practically useful.
• Trigonometric values are shown for the wrapped 0°–360° angle. If you enter 450°, the tool displays the trig values for 90°, which is the same physical direction.
• Precision is limited by double-precision floating-point arithmetic (about 15 significant decimal digits).
• The tool does not implement Bessel functions, spherical trigonometry or coordinate-system transforms. It handles scalar angle conversion only.
• The unit circle wraps angles outside 0°–360° for display. Negative angles are shown at their 360°-complement (for example, −45° is shown at 315°). The conversion itself uses the raw input value.
• The special-angle grid is limited to multiples of 15°. Angles like 22.5° or 7.5° are supported by direct input but are not shown in the grid.

Frequently asked questions

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