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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
°
° 57.29577951
Quick Values:
1 rad 5 rad 10 rad 25 rad 50 rad 100 rad 250 rad 500 rad 1,000 rad
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
Degree to Radian: Reciprocal Summary
Degree to Radian Conversion Reference Table
Reciprocal & Equivalent Value Scientific Notation
1 ° is equal to 0.0174533 rad (Radian) 1.745 × 10⁻²
5 ° is equal to 0.0872665 rad (Radian) 8.727 × 10⁻²
10 ° is equal to 0.174533 rad (Radian) 1.745 × 10⁻¹
50 ° is equal to 0.872665 rad (Radian) 8.727 × 10⁻¹
100 ° is equal to 1.745329 rad (Radian) 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

For an engineering input of 250 rad, applying the conversion factor yields exactly 14323.944878 °. At larger operational scale, 1000 Radian corresponds to 57295.779513 °.

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

°
SI Base Standard

A Degree (°) is the official base standard 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 (Base Standard)

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
Subtended angle of arc equal in length to circle radius (~57.3°)1 rad57.29578 °
Complete circular rotation angle expressed in radians (2π rad / 6.283 rad)6.283 rad359.989383 °
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].

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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.

A typical subtended angle of arc equal in length to circle radius (~57.3°) measures 1 rad, which corresponds to 57.2958 Degree (Degree).

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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