degrees to radians

Unit Converters

Instantly convert angles between Degrees and Radians with exact Pi fractions and trigonometric values.

Exact Pi Fraction

2Ï€3
2.094395 rad

Unit Circle (2D)

Trigonometric Context

sin(θ)√32
cos(θ)-12
tan(θ)-√3

How to Use the Degrees & Radians Converter

Master angle conversions, exact Pi fractions, unit circle geometry, and trigonometric ratios step by step.

1

Choose Input & Enter Value

Input your known angle in either Degrees (°) or Radians (rad). The bi-directional converter automatically calculates and updates the opposite field instantly in real time.

2

Flexible Pi & Fraction Notation

When entering radians, type mathematical notations such as 'pi', 'Ï€', fractions like '2pi/3', '3/4', or direct decimal numbers (e.g. 1.5708). The smart parser interprets all standard formats.

3

Inspect the Exact Pi Fraction

View the irreducible simplified exact fraction of π (e.g. 2π/3, π/6, 2π) along with a high-precision decimal representation up to 6 decimal places.

4

Visualize on the 2D Unit Circle

Watch the interactive coordinate circle update in real time. The rotating vector shows the exact terminal side, angle arc, quadrant position, and highlights standard benchmark angles.

5

Examine Trigonometric Ratios

Instantly check exact radical values and decimal approximations for sine, cosine, and tangent (sin θ, cos θ, tan θ) matching your angle.

6

Copy Mathematical Results

Click 'Copy Exact Math' to copy the complete formatted equation—including degrees, exact radian fraction, and trigonometric coordinates—directly to your clipboard.

Frequently Asked Questions

Technical insights and mathematical foundations of degrees and radians conversions.

Q:What is the formula to convert between degrees and radians?

The fundamental conversion relationship is 180° = π rad. To convert degrees to radians, multiply by π/180: rad = deg × (π / 180). To convert radians to degrees, multiply by 180/π: deg = rad × (180 / π) ≈ rad × 57.2958°.

Q:What is a radian and why is it preferred in calculus and physics?

A radian is the angle subtended at the center of a circle by an arc equal in length to the circle's radius (s = r · θ, so when arc length s = r, θ = 1 rad). Radians are the standard SI unit of angular measure because they allow calculus derivatives (like d/dx[sin x] = cos x) and physics rotational formulas (like v = ω · r) to work naturally without artificial conversion factors.

Q:How many degrees are in 1 radian (and radians in 1 degree)?

1 radian equals exactly 180° / π ≈ 57.29578° (approx. 57° 17' 45"). Conversely, 1 degree equals π / 180 ≈ 0.0174533 radians. A full circle of 360° corresponds to 2π ≈ 6.283185 radians.

Q:How do you simplify an angle into an exact fraction of π?

Divide the angle in degrees by 180 to form the fraction (degrees / 180), then reduce this fraction to its simplest irreducible form by dividing the numerator and denominator by their greatest common divisor (GCD). For example, 120° becomes 120/180, which simplifies by 60 to 2/3, yielding 2π/3 rad.

Q:What are the common standard angles on the unit circle?

Key reference angles include: 0° = 0 rad, 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, 90° = π/2 rad, 180° = π rad, 270° = 3π/2 rad, and 360° = 2π rad.

Q:Can this converter handle negative angles or values exceeding 360°?

Yes. Negative angles represent clockwise rotation, while angles greater than 360° (or 2π rad) represent multiple rotations. The converter accurately parses any real number, simplifies the exact π fraction, and renders the coterminal angle correctly on the unit circle.

Q:Why is the tangent of 90° (π/2) and 270° (3π/2) undefined (∞)?

Tangent is defined as the ratio of sine to cosine: tan(θ) = sin(θ) / cos(θ). At 90° and 270°, the cosine coordinate on the unit circle is 0 (cos 90° = 0). Division by zero is undefined in real arithmetic, creating vertical asymptotes where tangent approaches infinity (±∞).

r = d × (
Ï€ 180
)
  • r (Radians): The angular measurement in radians.
  • d (Degrees): The angular measurement in degrees.
  • Ï€ (Pi): Approximately 3.14159.

Degree to Radian Conversion Table

Degree [°]Radian [rad]
0.01 °0.0001745329 rad
0.1 °0.0017453293 rad
1 °0.0174532925 rad
2 °0.034906585 rad
3 °0.0523598776 rad
5 °0.0872664626 rad
10 °0.1745329252 rad
20 °0.3490658504 rad
50 °0.872664626 rad
100 °1.745329252 rad
1,000 °17.4532925199 rad

What is degrees to radians?

Instantly convert angles between Degrees and Radians with exact Pi fractions and trigonometric values. This calculator goes beyond simple decimals, giving you the precise mathematical output required for advanced geometry, physics, and programming. Toggle your input to instantly see the angle plotted on the interactive unit circle, providing vital spatial awareness. Furthermore, the engine automatically calculates the corresponding sine, cosine, and tangent ratios in pure fractional form, eliminating rounding errors from your workflow.
Practical Calculation Example
A mechanical engineer in Seattle is designing a custom cam mechanism and needs to program the CNC machine using precise radian measurements. They enter 120 into the DEGREES (°) field. The tool instantly outputs the EXACT PI FRACTION as 2π/3. By checking the TRIGONOMETRIC CONTEXT panel, they quickly verify the sin(θ) and cos(θ) ratios to determine the exact cutting vector before hitting Copy Exact Math to paste the pure formulas directly into their G-code script.
Reference Zones
Interface Element Mathematical Function Practical Application
RADIANS (RAD) Angular Measurement The standard unit of angular measure used in calculus and computer graphics.
EXACT PI FRACTION Algebraic Output Displays the unrounded, pure mathematical fraction (e.g., 2Ï€/3) alongside its decimal equivalent.
UNIT CIRCLE (2D) Spatial Visualization Plots the angle on a Cartesian coordinate system to instantly verify the vector's quadrant and direction.
TRIGONOMETRIC CONTEXT Ratio Calculation Provides the exact sine, cosine, and tangent values (e.g., √3/2) for the inputted angle.

History and Origin

The division of a circle into 360 degrees dates back to the ancient Babylonians, who used a sexagesimal (base-60) number system. The radian, however, is a much more recent concept, first proposed by the British mathematician Roger Cotes in 1714. He recognized that measuring angles by the arc length they subtend provides a more natural and mathematically elegant unit, which is exactly why radians are the absolute standard in modern calculus and physics today.

What is Degree?

Definition & Usage

A degree (symbol: °) is a measurement of a plane angle in which one full rotation is 360 degrees.

What is Radian?

Definition & Usage

The radian (symbol: rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.