Degrees to Radians Converter

The degrees to radians converter translates any angle between the two standard units of angular measurement used in mathematics, physics, and engineering. Enter a value in either Degrees to Radians Converter field and the other updates instantly. The conversion is exact, the factor π rad = 180° is the formal definition of the radian adopted by NIST and ISO 80000-3.

How to Use the Degrees to Radians Converter

1. Type a number into the Degrees (°) field to see the radian equivalent on the right.
2. Type a number into the Radians (rad) field to see the degree equivalent on the left.
3. The two Degrees to Radians Converter fields stay in sync as you type; editing one updates the other automatically. Use the quick presets for common reference points (30°, 45°, 60°, 90°, π, 2π, 1 rad, π/2 rad) to load an exact pair of values instantly.
4. Read the result box to see the active formula and a side-by-side display of the current value in both units.

The Conversion Formulas

The conversion between degrees and radians uses a single constant: the relationship π rad = 180°, which is the formal definition of the radian. Formally:

Degrees to radians: rad = deg × π / 180

Radians to degrees: deg = rad × 180 / π

The forward factor π / 180 ≈ 0.017453292519943295. The inverse factor 180 / π ≈ 57.29577951308232. These are exact up to the precision of π as represented in IEEE 754 double-precision floating-point arithmetic (about 15 significant decimal digits).

Why the factor π / 180? The radian is defined geometrically: one radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. The circumference of a full circle is 2π × radius, so a full turn corresponds to 2π radians. Since a full turn is also 360°, we have 360° = 2π rad, which simplifies to 180° = π rad, and hence 1° = π/180 rad. This relationship is exact by construction.

Why a degree = π / 180 rad and not some other factor? Because the radian is not an arbitrary unit, it is the natural, dimensionless ratio of arc length to radius. Any other scaling factor would not be dimensionless, and would break the elegant identity that sin(x) ≈ x for small x. The degree, by contrast, is an arbitrary historical division of the circle into 360 parts, inherited from the Babylonian base-60 number system.

Worked Examples

Example 1, A right angle. A geometry problem asks for the sine of a right angle. What is 90° in radians? rad = 90 × π / 180 = π / 2 ≈ 1.5707963 rad. So sin(90°) = sin(π/2) = 1 exactly.

Example 2, A 45° construction. A carpenter's square has a 45° angle. What is that in radians? rad = 45 × π / 180 = π / 4 ≈ 0.7853981 rad. So the carpenter's 45° angle is exactly π/4 rad, and sin(π/4) = cos(π/4) = √2 / 2 ≈ 0.7071.

Example 3, Half a turn. A pendulum swings through a half turn. Express the angle in radians. rad = 180 × π / 180 = π ≈ 3.1415927 rad. Half a turn is exactly π rad, regardless of the length of the pendulum or the value of g.

Example 4, Going the other way. A physics formula needs an angle of 1 radian. What is that in degrees? deg = 1 × 180 / π ≈ 57.2958°. So 1 rad ≈ 57.3°, slightly less than the 60° you'd guess from a regular hexagon.

Example 5, A full rotation. A wheel turns through one complete rotation. Express the angle in radians. rad = 360 × π / 180 = 2π ≈ 6.2832 rad. This is the conversion that makes angular velocity ω = θ / t and angular frequency ω = 2πf give consistent numerical answers.

Example 6, A frequency problem. A motor spins at 60 revolutions per minute. Express the angular frequency in rad/s. Each revolution is 2π rad, so ω = 60 rev/min × 2π rad/rev × 1 min / 60 s = 2π rad/s ≈ 6.2832 rad/s. The conversion factor 2π is the same one used in the degrees-to-radians formula, applied at the unit level.

Where the Conversion Shows Up

• Trigonometry and pre-calculus: Converting between degrees (used in tables, textbooks, and most calculator displays) and radians (used in formulas). The identities sin(π/6) = 1/2, cos(π/4) = √2/2, tan(π/3) = √3 are easier to memorise in radians.
• Calculus: Derivatives of sin, cos, and tan, d/dx sin(x) = cos(x), d/dx cos(x) = −sin(x), hold only when x is in radians. The same is true for Taylor series sin(x) = x − x³/6 + ... and small-angle approximations sin(x) ≈ x.
• Physics: Angular frequency ω = 2πf, rotational kinetic energy KE = ½Iω², simple harmonic motion x(t) = A cos(ωt), wave number k = 2π/λ, all expect radians. The factor 2π appears because a full cycle is 2π rad.
• Engineering: Fourier analysis, signal processing, control systems, robotics, 3D graphics (rotations around an axis), and game physics all use radians internally. Computer graphics libraries like OpenGL and DirectX take rotation arguments in radians.
• Surveying and navigation: Compass bearings and theodolite readings are reported in degrees (or grads, where a full circle is 400 grads) for human consumption, but the underlying sensors and computations work in radians.
• Astronomy: Celestial coordinates are typically given in degrees, arcminutes, and arcseconds (1° = 60′ = 3600″). For very high precision (parallax measurements, exoplanet detection), angles may be expressed in radians or milliarcseconds.
• Education: Teaching the difference between arbitrary units (degrees) and natural, dimensionless units (radians) is a key conceptual step in moving from school mathematics to university-level physics and engineering.

Common Mistakes

• Using degrees inside a radian-only formula. The Taylor series sin(x) = x − x³/6 + ... gives nonsense if x is in degrees. Plugging x = 90° gives sin(90°) ≈ 90 − 90³/6 ≈ −121,335, which is wildly wrong. Always convert to radians before using calculus identities.
• Forgetting the factor of π. A common error is to claim that "180° = 180 rad" because the number is the same. It is not, a right angle is 90° = π/2 rad ≈ 1.5708 rad, not 90 rad. The number alone is meaningless without the unit.
• Rounding π too early. If you write π ≈ 3.14 and then convert 30° to radians, you get 30 × 3.14 / 180 ≈ 0.5233 rad. The exact value is 30 × π / 180 = π/6 ≈ 0.5235987... rad. The difference (about 0.05%) is small, but for chained calculations (especially when the result feeds into a derivative or integral) it can accumulate.
• Mixing degrees and radians in the same formula. If you write ω = θ × f, make sure θ is in radians and f is in Hz. Mixing units gives a number that is off by a factor of 360 or 2π.
• Confusing grads with degrees. A few scientific calculators and surveying tools use grads (also called gradians or gon), where a full circle = 400 grad. Make sure your calculator is in DEG mode, not GRAD, before reading an angle.
• Forgetting to convert when calling math libraries. In JavaScript, Math.sin(Math.PI / 4) returns 0.7071... (correct). Math.sin(45) returns 0.8509... (wrong, because the argument is interpreted as 45 radians). In Python, numpy.sin(np.deg2rad(45)) is the safe pattern.
• Treating radians as a unit with a fixed symbol. The radian is dimensionless (it is a ratio of two lengths), so it is written as just "rad" or sometimes omitted entirely in advanced texts. The SI explicitly treats the radian as a special dimensionless derived unit.

Quick Mental Conversion Trick

For an approximate deg → rad conversion in your head: multiply the degree value by 0.01745 (or equivalently, divide by 57.3). For example, 60° → 60 × 0.01745 ≈ 1.047 rad (actual: π/3 ≈ 1.0472 rad, very close). 120° → 120 × 0.01745 ≈ 2.094 rad (actual: 2π/3 ≈ 2.0944 rad).

For rad → deg: multiply by 57.3. For example, 2 rad → 2 × 57.3 ≈ 114.6° (actual: 114.59°). 3 rad → 3 × 57.3 ≈ 171.9° (actual: 171.89°).

The trick works because π/180 ≈ 0.017453 and 180/π ≈ 57.2958. The error introduced by using 0.01745 instead of the exact factor is about 0.02%, which is far smaller than the precision you can read off a mental calculation.

A more memorable sanity check: 1 radian is just under 60° (specifically about 57.3°), and 2 radians is just over 114° (specifically about 114.6°). If you remember these two anchors, you can interpolate most other angles in your head.

Frequently Asked Questions

Q: What is the formula to convert degrees to radians? A: Use rad = deg × π / 180. The factor is approximately 0.017453292519943295. For example, 90° × π/180 = π/2 ≈ 1.5707963 rad, and 180° × π/180 = π ≈ 3.1415927 rad. The constant π / 180 follows directly from the definition of the radian: one radian is the angle subtended by an arc equal to the radius, and 180° of arc equals π radians of angle by the geometry of a semicircle.

Q: What is the formula to convert radians to degrees? A: Use deg = rad × 180 / π. The factor is approximately 57.29577951308232. For example, 1 rad × 180/π ≈ 57.2958°, and π rad × 180/π = 180° exactly. Going the other way is sometimes more intuitive: 1 radian is just under 60°, and 2 radians is just over 114°.

Q: Why do calculators default to degrees instead of radians? A: Most consumer calculators default to degrees because degrees are the natural unit for everyday navigation, surveying, and most school mathematics. Scientific calculators include a MODE or DRG button that toggles between DEG (degrees), RAD (radians), and GRAD (gradians, where a full circle is 400). For pure-mathematics, physics, and engineering work, switch to RAD mode because the formulas d/dx sin(x) = cos(x) and sin(x) ≈ x (small x) are only correct when x is in radians.

Q: What is the difference between radians and degrees? A: A degree is an arbitrary division of the full circle into 360 equal parts, originally chosen by the Babylonians in base-60 arithmetic. A radian is defined geometrically: one radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. A full circle is 2π radians, so π radians = 180°. Radians are dimensionless, they are a ratio of two lengths, which makes them the natural unit for trigonometric series, derivatives, and physics formulas involving angular quantities.

Q: When should I use radians instead of degrees? A: Use radians in any formula or computation that involves a trigonometric function whose argument needs a derivative, integral, or limit. This includes: derivatives of sin and cos (d/dx sin(x) = cos(x)), small-angle approximations (sin(x) ≈ x, tan(x) ≈ x for small x), Taylor series (sin(x) = x − x³/6 + ...), physics formulas for angular frequency ω (ω = 2πf), rotational kinetic energy (½Iω²), Fourier analysis, and complex exponentials (e^{iθ}). Use degrees for navigation bearings, surveying angles, geographic coordinates, and everyday display in countries that use the degree system (which is most of them).

Q: How are radians used in surveying and angles of elevation? A: Surveyors and civil engineers usually measure horizontal and vertical angles in degrees (or in grads, where a full circle = 400 grads), because the angles are reported to humans and the precision of one degree (or one minute of arc, 1/60 of a degree) is appropriate for most practical work. Internally, however, theodolites and total stations compute in radians, the chip measures angles as arc length divided by radius, then converts to degrees for the operator's display. For very precise astronomical or geodetic work, angles may be quoted in arcseconds (1/3600 of a degree), milliarcseconds, or radians directly.

Q: Can this calculator be used for trigonometry, physics, or engineering problems? A: Yes. The conversion factors are exact by the definition of the radian and are appropriate for trigonometry, calculus, physics (angular frequency, rotational motion, wave equations), and engineering (signal processing, robotics, control systems). For high-stakes applications (aerospace, medical imaging, surveying near legal boundaries), verify results with a domain expert. The formulas used are well-established and validated against the NIST and ISO 80000-3 definitions of the radian.

References

• NIST Special Publication 811: Guide for the Use of the SI Units, defines the radian as a dimensionless derived SI unit and confirms the relationship π rad = 180°.
• ISO 80000-3: Quantities and units, Part 3: Space and time, international standard defining plane angle and solid angle, including the radian and steradian.
• BIPM SI Brochure (9th edition, 2019), the formal definition of the radian as the angle for which the arc length equals the radius.
• International Electrotechnical Commission (IEC) 80000-6, defines the hertz and angular frequency in radians per second.
• Abramowitz, M. and Stegun, I. A. (eds.), Handbook of Mathematical Functions, standard reference for trigonometric series in radians.