Understanding the Conversion of Angular Distance to Terrestrial Distance Using Earth’s Radius

Angular distance conversion translates an angle measured in degrees into a physical distance on Earth’s surface. This calculation is essential for navigation, geodesy, and mapping applications.

By leveraging Earth’s radius, this article details formulas, tables, and real-world examples to accurately convert angular measurements into terrestrial distances.

• Calculate terrestrial distance for 15° angular separation using Earth’s radius.
• Convert 0.5 radians angular distance to kilometers on Earth’s surface.
• Find the distance between two points separated by 30 arcminutes on Earth.
• Determine terrestrial distance for 1° angular distance at Earth’s equator.

Comprehensive Tables of Angular to Terrestrial Distance Conversion

Below are detailed tables showing the conversion of common angular distances into terrestrial distances using the average Earth radius of 6,371 kilometers. These values are crucial for quick reference in geospatial calculations.

Note: The terrestrial distances are calculated assuming a perfect sphere Earth model with radius 6,371 km. Actual distances may vary slightly due to Earth’s oblateness.

Mathematical Formulas for Angular to Terrestrial Distance Conversion

Converting angular distance to terrestrial distance involves fundamental geometric relationships on a sphere. The Earth is approximated as a sphere with radius R, and the angular distance between two points is denoted as θ.

Basic Arc Length Formula

The terrestrial distance d corresponding to an angular distance θ (in radians) on a sphere of radius R is given by the arc length formula:

• d: Terrestrial distance along the Earth’s surface (kilometers or miles)
• R: Earth’s radius (average 6,371 km or 3,959 miles)
• θ: Angular distance in radians

Since angular distances are often given in degrees, conversion to radians is necessary:

Conversion of Angular Distance in Degrees to Terrestrial Distance

Combining the two formulas, the terrestrial distance d can be expressed as:

This formula is the most commonly used for converting angular distances measured in degrees to terrestrial distances.

Angular Distance in Arcminutes and Arcseconds

For finer angular measurements, such as arcminutes (‘) and arcseconds (“), the conversion to degrees is:

• 1 degree = 60 arcminutes
• 1 arcminute = 60 arcseconds

Thus, to convert arcminutes or arcseconds to radians:

or

Great-Circle Distance Formula

When calculating the shortest distance between two points on Earth’s surface given their latitudes and longitudes, the angular distance Δσ is derived from the spherical law of cosines:

• φ₁, φ₂: Latitudes of point 1 and point 2 in radians
• Δλ: Difference in longitudes (λ₂ – λ₁) in radians
• Δσ: Central angle between the two points in radians

Once Δσ is calculated, the terrestrial distance d is:

Haversine Formula for Angular Distance

For improved numerical stability, especially for small distances, the haversine formula is preferred:

• Δφ: Difference in latitudes (φ₂ – φ₁) in radians
• Δλ: Difference in longitudes (λ₂ – λ₁) in radians

Then, terrestrial distance is:

Detailed Explanation of Variables and Common Values

• Earth’s Radius (R): The average radius is 6,371 km (3,959 miles). Earth is not a perfect sphere but an oblate spheroid, so radius varies slightly:
  • Equatorial radius ≈ 6,378.1 km
  • Polar radius ≈ 6,356.8 km
• Angular Distance (θ or Δσ): The central angle between two points on Earth’s surface, measured in degrees or radians.
• Latitude (φ): Angular distance north or south of the equator, ranges from -90° to +90°.
• Longitude (λ): Angular distance east or west of the Prime Meridian, ranges from -180° to +180°.

For most practical applications, the average Earth radius (6,371 km) suffices. However, for high-precision geodesy, the WGS84 ellipsoid model is used, which accounts for Earth’s flattening.

Real-World Applications and Examples

Example 1: Calculating Distance Between Two Cities Using Angular Distance

Calculate the terrestrial distance between New York City (40.7128° N, 74.0060° W) and London (51.5074° N, 0.1278° W) using the spherical law of cosines.

• Convert latitudes and longitudes to radians:
  • φ₁ = 40.7128 × π / 180 = 0.7106 rad
  • λ₁ = -74.0060 × π / 180 = -1.2916 rad
  • φ₂ = 51.5074 × π / 180 = 0.8990 rad
  • λ₂ = -0.1278 × π / 180 = -0.0022 rad
• Calculate Δλ:
  • Δλ = λ₂ – λ₁ = -0.0022 – (-1.2916) = 1.2894 rad
• Apply spherical law of cosines:
  • cos(Δσ) = sin(0.7106) × sin(0.8990) + cos(0.7106) × cos(0.8990) × cos(1.2894)
  • cos(Δσ) ≈ 0.6523 × 0.7826 + 0.7579 × 0.6225 × 0.2780 = 0.5103 + 0.1313 = 0.6416
• Calculate Δσ:
  • Δσ = arccos(0.6416) ≈ 0.8703 rad
• Calculate terrestrial distance:
  • d = 6,371 km × 0.8703 = 5,546 km

This result aligns closely with the known great-circle distance between New York and London.

Example 2: Determining Distance for a Given Angular Separation at the Equator

Find the terrestrial distance corresponding to an angular distance of 1.5° along the Earth’s equator.

• Convert 1.5° to radians:
  • θ = 1.5 × π / 180 = 0.02618 rad
• Calculate distance:
  • d = 6,371 km × 0.02618 = 166.8 km

This means traveling 1.5° along the equator corresponds to approximately 167 kilometers.

Additional Considerations for Precision and Practical Use

While the spherical Earth model simplifies calculations, real-world applications often require adjustments for Earth’s ellipsoidal shape. The WGS84 ellipsoid parameters are:

• Equatorial radius (a): 6,378.137 km
• Polar radius (b): 6,356.752 km
• Flattening (f): 1/298.257223563

For high-precision geodesy, Vincenty’s formulae or other ellipsoidal distance calculations are preferred. However, for most navigation and mapping tasks, the spherical model and formulas presented here provide sufficient accuracy.

Additionally, when converting angular distances to terrestrial distances, consider:

• Latitude dependence: The length of a degree of longitude varies with latitude, shrinking to zero at the poles.
• Altitude effects: Distances measured at altitude differ slightly due to increased radius.
• Measurement units: Always ensure angular units are consistent (degrees vs radians) and convert accordingly.

Summary of Key Formulas

Recommended External Resources for Further Study

• NOAA Technical Report on Geodesic Calculations
• Great-circle distance – Wikipedia
• ICSN Geodesy Standards and Earth Models
• NOAA Geodesic Inverse and Forward Calculations Tool