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Degree to Radian (° to rad) Calculator & Mathematical Formula

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Live Conversion • ISO 80000-3:2019 Certified
0.1 Degree = 0.0017453293 Radian
1 ° = 0.0174533 rad • 1 rad = 57.29578 °
Degree to Radian: One Degree is equal to 0.017453293 Radians.
Certified Authority Value
0.00174533
rad
Exact: 1 rad ≡ 57.2957795131 °
°
°
Swap to Radian to Degree
Swap to Radian to Degree
rad
rad 0.0017453293
Quick Values:
Conversion Formula
To convert Degree to Radian, multiply by 0.017453293
° × 0.017453293 = rad

Calculation: 0.1 ° × 0.017453293 = 0.0017453293 rad

Common Degree to Radian Calculations

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

Tap any row to calculate
Degree to Radian: Primary Direct Table (15 Steps)
Degree to Radian Conversion Reference Table
Conversion & Equivalent Value Scientific Notation
0.001 ° 1.745329 × 10⁻⁵ rad 1.745 × 10⁻⁵
0.01 ° 0.000174533 rad 1.745 × 10⁻⁴
0.1 ° 0.00174533 rad 1.745 × 10⁻³
0.5 ° 0.00872665 rad 8.727 × 10⁻³
1 ° 0.0174533 rad 1.745 × 10⁻²
2 ° 0.0349066 rad 3.491 × 10⁻²
5 ° 0.0872665 rad 8.727 × 10⁻²
10 ° 0.174533 rad 1.745 × 10⁻¹
20 ° 0.349066 rad 3.491 × 10⁻¹
25 ° 0.436332 rad 4.363 × 10⁻¹
50 ° 0.872665 rad 8.727 × 10⁻¹
100 ° 1.745329 rad 1.745 × 10⁰
250 ° 4.363323 rad 4.363 × 10⁰
500 ° 8.726646 rad 8.727 × 10⁰
1000 ° 17.453293 rad 1.745 × 10¹
Radian to Degree: Reciprocal Summary
Radian to Degree Conversion Reference Table
Reciprocal & Equivalent Value Scientific Notation
1 rad 57.29578 ° 5.73 × 10¹
5 rad 286.478898 ° 2.865 × 10²
10 rad 572.957795 ° 5.73 × 10²
50 rad 2864.788976 ° 2.865 × 10³
100 rad 5729.577951 ° 5.73 × 10³
View Complete 15-Row Radian to Degree 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 Degree (°) and Radian (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 °

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

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

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

How to Convert Degree to Radian

Converting Degree (°) to Radian (rad) is a core calculation in Plane Angle, which measures the circular arc subtended between two intersecting rays or diverging lines. Because Radian is a broader physical scale than Degree, converting from ° to rad condenses the numerical figure. It takes 57.29578 Degree to equal a single Radian (1 ° = 0.0174533 rad). To convert any quantity, multiply by 0.0174533 (or divide by 57.29578).

Direct Factor1 ° = 0.0174533 rad
Inverse Ratio1 rad = 57.29578 °
Conversion ScaleDirect Proportional Magnitude Compression
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 ° = 1 radian (rad)
2. Target SI Bridge (ktarget):1 rad = 57.29578 radian (rad)
Step-by-step substitution for Degree → Radian:V(rad) = V(°) × ( 1 / 57.29578 ) = V(°) × 0.0174533
Mathematical FormulaValue in Radian = Value in Degree × 0.017453293Direct computational formula applied for standard conversion.
Worked Example

Evaluating an input of 50 Degree (°) produces 0.872665 Radian (rad). In heavy-duty utility and industrial workloads, 240 ° equals 4.18879 rad.

Inverse check: 1 rad = 57.29578 °.

Understanding Degree and Radian

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

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

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 Radian
Perpendicular orthogonal intersection right angle (90°)90 °1.570796 rad
Straight line flat angle rotation in Euclidean plane geometry (180°)180 °3.141593 rad
Complete 360-degree full circle revolution rotation (360°)360 °6.283185 rad
Equilateral triangle internal corner interior angle (60°)60 °1.047198 rad
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:
0.1 ° at 28" Commuter Bicycle Wheel (0.35 m) = 0.0006109 m
Equals 6.1087 × 10⁻⁷ km • 0.06109 cm • 0.6109 mm
SI Length
0.0006109
m
Meters
Geographic Scale
6.1087 × 10⁻⁷
km
Kilometers
Engineering Scale
0.06109
cm
Centimeters
Precision Machining
0.6109
mm
Millimeters
US Architectural
0.002004
ft
Feet
US Tooling
0.02405
in
Inches
  1. 1. Normalize Input to SI Base Unit (°):
    θ = 0.1 ° × 1 = 0.1 °
  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) × (0.1 °) = 0.00061087 m
  4. 4. Convert to Standard Engineering Units:
    • m: 0.00061087 × 1 = 0.0006109 m (Meters) • km: 0.00061087 × 0.001 = 6.1087 × 10⁻⁷ km (Kilometers) • cm: 0.00061087 × 100 = 0.06109 cm (Centimeters) • mm: 0.00061087 × 1000 = 0.6109 mm (Millimeters) • ft: 0.00061087 × 3.2808 = 0.002004 ft (Feet) • in: 0.00061087 × 39.3701 = 0.02405 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

The mathematical formula to convert degrees to radians is: [value] * 0.017453293 = radians. 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 radians back to degrees, you must divide your starting value by 0.017453293. Our bidirectional tool automatically computes this inverse equation the moment you type into the radian input field.

1 Radian is larger than 1 Degree. Specifically, 1 Degree equals 0.017453293 radians, meaning 1 Radian is 57.2958 times larger than 1 Degree (a difference of 1.76 orders of magnitude). Expressing measurements in degrees requires a larger numerical value to represent the equivalent physical magnitude.

Under standard international measurement guidelines, 1 Degree converts to 0.017453293 radians. Precise numerical conversion factors ensure consistency across laboratory, engineering, and field data.

There are 0.0174533 radians in a single degree. In technical metrology notation: 1 ° = 0.0174533 rad.

To calculate radians from degrees, multiply the degree value by 0.0174533. The formula is: V(rad) = V(°) × 0.0174533.

To convert from radians to degrees, multiply the radian value by 57.2958. The inverse formula is: V(°) = V(rad) × 57.2958.

Radians are the standard SI unit for angular measure, defining an angle by the ratio of an arc's length to its radius. They are universally used in calculus and harmonic physics.

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