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Can U Calculate Arc Distance N to S

Reviewed by Calculator Editorial Team

Calculating arc distance between two points on a sphere (like Earth) is essential for navigation, geography, and physics. This guide explains how to compute the shortest distance between two points using their latitude and longitude coordinates.

What is Arc Distance?

Arc distance, also known as great-circle distance, is the shortest distance between two points on the surface of a sphere. On Earth, this is the path followed by an airplane or ship traveling between two cities.

Unlike straight-line (Euclidean) distance, arc distance accounts for the curvature of the Earth. For example, the distance between New York and London isn't a straight line but follows the curve of the planet.

How to Calculate Arc Distance

To calculate arc distance, you need:

  • The latitude and longitude of both points in decimal degrees
  • The radius of the sphere (Earth's radius is approximately 6,371 km)

The calculation involves converting coordinates to radians, applying the haversine formula, and then multiplying by the radius to get the distance in kilometers or miles.

The Formula

a = sin²(Δφ/2) + cos(φ1) * cos(φ2) * sin²(Δλ/2) c = 2 * atan2(√a, √(1−a)) d = R * c

Where:

  • φ1, φ2 = latitudes of points 1 and 2 in radians
  • Δφ = φ2 - φ1
  • Δλ = λ2 - λ1 (difference in longitudes in radians)
  • R = Earth's radius (6,371 km or 3,959 miles)

Worked Example

Example Calculation

Calculate the distance between:

  • Point 1: 40.7128° N, 74.0060° W (New York)
  • Point 2: 51.5074° N, 0.1278° W (London)

Using the formula and Earth's radius of 6,371 km, the arc distance is approximately 5,570 km.

FAQ

What units should I use for latitude and longitude?

Use decimal degrees (e.g., 40.7128° for latitude). Degrees, minutes, and seconds can be converted to decimal degrees.

Can I use this formula for other planets?

Yes, replace Earth's radius with the planet's equatorial radius. For Mars, use approximately 3,390 km.

What if my points are in the southern hemisphere?

The formula works for all hemispheres. Negative values indicate south latitude or west longitude.