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Radian to Degree (rad to °) Calculator & Formula

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Live Conversion • ISO 80000-3:2019 Certified
1 Radian = 57.29577951 Degree
1 rad = 57.29578 ° • 1 ° = 0.0174533 rad
Radian to Degree: One Radian is equal to 57.29577951 Degrees.
Certified Authority Value
57.29577951
°
Exact: 1 ° ≡ 0.01745329252 rad
rad
rad
Swap to Degree to Radian
Swap to Degree to Radian
°
° 57.29577951
Quick Values:
Conversion Formula
To convert Radian to Degree, multiply by 57.29577951
rad × 57.29577951 = °

Calculation: 1 rad × 57.29577951 = 57.29577951 °

Common Radian to Degree Calculations

Quick-lookup reference table showing calculated equivalents for standard increments between Radian (rad) and Degree (°).

Tap any row to calculate
Radian to Degree: Primary Direct Table (15 Steps)
Radian to Degree Conversion Reference Table
Conversion & Equivalent Value Scientific Notation
0.001 rad 0.0572958 ° 5.73 × 10⁻²
0.01 rad 0.572958 ° 5.73 × 10⁻¹
0.1 rad 5.729578 ° 5.73 × 10⁰
0.5 rad 28.64789 ° 2.865 × 10¹
1 rad 57.29578 ° 5.73 × 10¹
2 rad 114.591559 ° 1.146 × 10²
5 rad 286.478898 ° 2.865 × 10²
10 rad 572.957795 ° 5.73 × 10²
20 rad 1145.91559 ° 1.146 × 10³
25 rad 1432.394488 ° 1.432 × 10³
50 rad 2864.788976 ° 2.865 × 10³
100 rad 5729.577951 ° 5.73 × 10³
250 rad 14323.944878 ° 1.432 × 10⁴
500 rad 28647.889757 ° 2.865 × 10⁴
1000 rad 57295.779513 ° 5.73 × 10⁴
Degree to Radian: Reciprocal Summary
Degree to Radian Conversion Reference Table
Reciprocal & Equivalent Value Scientific Notation
1 ° 0.0174533 rad 1.745 × 10⁻²
5 ° 0.0872665 rad 8.727 × 10⁻²
10 ° 0.174533 rad 1.745 × 10⁻¹
50 ° 0.872665 rad 8.727 × 10⁻¹
100 ° 1.745329 rad 1.745 × 10⁰
View Complete 15-Row Degree to Radian Table

Real-World Context & Everyday Benchmarks

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To help contextualize these units in everyday life, science, and engineering, here are verified real-world reference points for Radian (rad) and Degree (°):

Radian References:

  • Subtended angle of arc equal in length to circle radius (~57.3°) 1 rad
  • Complete circular rotation angle expressed in radians (2π rad / 6.283 rad) 6.283 rad

Degree References:

  • Perpendicular orthogonal intersection right angle (90°) 90 °
  • Straight line flat angle rotation in Euclidean plane geometry (180°) 180 °
  • Complete 360-degree full circle revolution rotation (360°) 360 °
  • Equilateral triangle internal corner interior angle (60°) 60 °

Tap any benchmark above to automatically calculate its exact equivalent in Degree.

ISQ Dimension: Dimensionless [L / L] Coherent SI Unit: radian (rad) Geoscience, Geodesy & Petroleum Engineering

How to Convert Radian to Degree

Converting Radian (rad) to Degree (°) is a core calculation in Plane Angle, which measures the circular arc subtended between two intersecting rays or diverging lines. Because Radian represents a larger magnitude than Degree, converting from rad to ° expands the numerical figure. Exactly 1 Radian encompasses 57.29578 Degree. To convert any quantity, multiply the value in Radian by 57.29578.

Direct Factor1 rad = 57.29578 °
Inverse Ratio1 ° = 0.0174533 rad
Conversion ScaleDirect Proportional Magnitude Expansion
First-Principles SI Derivation ProofNIST SP 811 / ISO 80000 Compliance

Mathematical Conversion Derivation via Coherent SI Base Unit

Under the International System of Quantities (ISQ), dimensional conversion between non-SI units is rigorously resolved by bridging through the coherent SI base unit (radian (rad)) without arbitrary rounding:

Vtarget=Vsource×(ksource→SIktarget→SI)V_{\text{target}} = V_{\text{source}} \times \left( \frac{k_{\text{source} \to \text{SI}}}{k_{\text{target} \to \text{SI}}} \right)
Universal Dimensional Conversion Law (ISO 80000 / NIST SP 811)
1. Source SI Bridge (ksource):1 rad = 57.29578 radian (rad)
2. Target SI Bridge (ktarget):1 ° = 1 radian (rad)
Step-by-step substitution for Radian → Degree:V(°) = V(rad) × ( 57.29578 / 1 ) = V(rad) × 57.29578
Mathematical FormulaValue in Degree = Value in Radian × 57.29577951Direct computational formula applied for standard conversion.
Worked Example

To convert 24 Radian (rad) to Degree (°): multiply 24 by 57.29578 to get 1375.098708 °. At an industrial engineering scale, 100 rad converts to 5729.577951 °.

Inverse check: 1 ° = 0.0174533 rad.

Understanding Radian and Degree

Radian

rad
Standard Unit

A Radian (rad) is a recognized unit for plane angle.

Applications & Practical Context

In practical angle terms, a subtended angle of arc equal in length to circle radius (~57.3°) equals 1 rad, a complete circular rotation angle expressed in radians (2π rad / 6.283 rad) equals 6.283 rad.

Physical Setup

A physical magnitude of 1 rad corresponds to 57.29578 rad in standard SI measurement.

Conversion to SI Unit: radian (rad)
1 rad = 57.29578 rad

Degree

°
Standard Unit

A Degree (°) is a recognized unit for plane angle.

Applications & Practical Context

In practical angle terms, a perpendicular orthogonal intersection right angle (90°) equals 90 °, a straight line flat angle rotation in Euclidean plane geometry (180°) equals 180 °, a complete 360-degree full circle revolution rotation (360°) equals 360 °.

Physical Setup

A plane angle of 1 ° represents the angular divergence subtending an arc equal to 1 Degree.

Conversion to SI Unit: radian (rad)
1 ° = 1 rad

Domain Metrology & Engineering Applications

Governing Law & Equation

Subtended Circular Arc Ratio (θ = s / r)

Measurement Instrumentation

Rigorously quantified in laboratory and field settings using electronic digital theodolites, optical polygon autocollimators, and rotary optical angle encoders.

Primary Industry Sectors

Applied across geodetic triangulation surveys, aerospace satellite gimbal tracking, civil bridge alignment, and telescope mount tracking under international metrology standard ISO 80000-3:2019.

Physical Orders of Magnitude & Laboratory Reference Benchmarks

Physical Phenomenon / Engineering ReferenceReference ValueConverted to Degree
Perpendicular orthogonal intersection right angle (90°)90 °5156.620156 °
Straight line flat angle rotation in Euclidean plane geometry (180°)180 °10313.240312 °
Complete 360-degree full circle revolution rotation (360°)360 °20626.480625 °
Equilateral triangle internal corner interior angle (60°)60 °3437.746771 °
Institutional Metrology & Standard Authority:

Formulas, conversion coefficients, and base definitions on this page strictly adhere to the ISO 80000-3:2019 and the BIPM SI Brochure (9th Edition). Governed by Subtended Circular Arc Ratio (θ = s / r) under ISO 80000 dimensional vector Dimensionless [L / L].

Multi-Variable Engineering Bridge Trigonometry & Circular Trajectory

Angular Rotation to Tangential Arc Length Bridge (Meters • Centimeters • Millimeters • Kilometers • Feet • Inches)

Calculate curved perimeter arc length s = r × θ × (π/180) subtended along circular trajectories, bridging angular rotation and travel distance [L] via radius (r).

s = r × θ × (π/180)
Vector: [°] × [π/180 rad/°] × [m] → [m]
Live Calculated Bridge Equivalent:
1 rad at 28" Commuter Bicycle Wheel (0.35 m) = 0.35 m
Equals 0.00035 km • 35 cm • 350 mm
SI Length
0.35
m
Meters
Geographic Scale
0.00035
km
Kilometers
Engineering Scale
35
cm
Centimeters
Precision Machining
350
mm
Millimeters
US Architectural
1.1483
ft
Feet
US Tooling
13.7795
in
Inches
  1. 1. Normalize Input to SI Base Unit (°):
    θ = 1 rad × 57.29577951 = 57.29577951 °
  2. 2. Apply Engineering Parameter (r):
    Selected condition: 28" Commuter Bicycle Wheel (0.35 m) at r = 0.35 m
  3. 3. Execute Governing Physical Law in SI Units:
    s = r × θ × (π/180) = (0.01745 rad/°) × (0.35 m) × (57.29577951 °) = 0.35 m
  4. 4. Convert to Standard Engineering Units:
    • m: 0.35 × 1 = 0.35 m (Meters) • km: 0.35 × 0.001 = 0.00035 km (Kilometers) • cm: 0.35 × 100 = 35 cm (Centimeters) • mm: 0.35 × 1000 = 350 mm (Millimeters) • ft: 0.35 × 3.2808 = 1.1483 ft (Feet) • in: 0.35 × 39.3701 = 13.7795 in (Inches)
Explore Arc Length Converter →
Engineering Substance / Operating Standard Parameter Value (m) Reference State / Condition Typical Industrial & Engineering Application
Wall Clock Minute Hand (0.15 m) 0.15 0.15 m (150 mm) Mechanical analog clock tip trajectory radius (15 cm)
28" Commuter Bicycle Wheel (0.35 m) 0.35 0.35 m (350 mm) Standard 700c tire tread rolling radius
Utility Wind Turbine Blade (60 m) 60 60.0 meters Offshore 4 MW turbine blade tip rotation perimeter
Earth Equatorial Radius (6,378 km) 6378137 6,378,137 meters WGS 84 terrestrial equatorial geographic radius
Geostationary Orbital Radius (42,164 km) 4.2164 × 10⁷ 42,164,000 meters Satellite geosynchronous orbital perimeter radius
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Frequently Asked Questions

There are exactly 57.29577951 degrees (°) in 1 radian (rad). To calculate larger amounts manually, simply multiply your total number of radians by the conversion multiplier of 57.29577951.

The mathematical formula to convert radians to degrees is: [value] * 57.29577951 = degrees. Rather than doing the math manually, you can use our high-precision online calculator to process this exact equation instantly without rounding errors.

To perform the reverse calculation from degrees back to radians, you must divide your starting value by 57.29577951. Our bidirectional tool automatically computes this inverse equation the moment you type into the degree input field.

1 Radian is larger than 1 Degree. Specifically, 1 Radian equals exactly 57.29577951 degrees, making it 57.2958 times larger (a difference of 1.76 orders of magnitude). Therefore, quantities expressed in degrees require a larger numerical value to represent the same physical amount.

Under standard international measurement guidelines, 1 Radian converts to 57.29577951 degrees. Precise numerical conversion factors ensure consistency across laboratory, engineering, and field data.

Exactly 1 rad = 57.2957795131 ° (commonly rounded to 57.2958 °). This derives from the definition: a full circle is 2π radians = 360 °, so 1 rad = 360 / (2π) = 180/π ≈ 57.2958 °.

Multiply the radian value by 180/π (or 57.2957795131). KaTeX: V(°) = V(rad) × 180 / π ≈ V(rad) × 57.2958. Example: π/2 rad × 57.2958 = 90 °.

Multiply the degree value by π/180 (or 0.01745329252). KaTeX: V(rad) = V(°) × π / 180 ≈ V(°) × 0.0174533. Example: 45 ° × 0.0174533 = 0.785398 rad = π/4.

1 radian ≈ 57.3 ° — roughly the angle subtended by an arc equal to the radius. In mechanical engineering, a 1 rad shaft rotation moves a point at 1 m radius by 1 m arc length. In optics, 1 rad ≈ 57.3 ° beam divergence.

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