Maths⏱ 5 min read
How to Calculate the Distance Between Two Points
The distance formula in 2D and 3D comes directly from Pythagoras. Here is how it works, how GPS uses it, and how to calculate great-circle distance on the curved surface of the Earth.
The distance between two points is a fundamental calculation in mathematics with real-world applications from GPS routing to physics simulations. Understanding the derivation makes it unforgettable.
2D Distance Formula
Distance = sqrt((x2-x1)^2 + (y2-y1)^2)
This is simply Pythagoras applied to the horizontal and vertical
separation between two points.
Example: Point A at (2, 3), Point B at (7, 9)
Horizontal separation (Δx): 7 - 2 = 5
Vertical separation (Δy): 9 - 3 = 6
Distance = sqrt(5^2 + 6^2) = sqrt(25 + 36) = sqrt(61) = 7.81 units
Why this works: draw a right triangle with the two points
at opposite corners. Pythagoras gives the hypotenuse.
3D Distance Formula
Distance = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
Just adds the vertical (z) dimension.
Example: Point A at (1, 2, 3), Point B at (4, 6, 3)
Δx = 3, Δy = 4, Δz = 0
Distance = sqrt(9 + 16 + 0) = sqrt(25) = 5 units
Example with height:
Building at (0, 0, 0), drone at (100, 50, 30)
Distance = sqrt(10000 + 2500 + 900) = sqrt(13400) = 115.8m
Manhattan Distance (City Block Distance)
In grid-based navigation (like city streets), you cannot travel
diagonally — only horizontally and vertically.
Manhattan distance = |x2-x1| + |y2-y1|
Same example (2,3) to (7,9):
Manhattan = |7-2| + |9-3| = 5 + 6 = 11 units
vs Euclidean: 7.81 units
Manhattan distance is used in:
- City navigation routing
- Machine learning (k-nearest neighbours)
- Some computer vision algorithms
Great-Circle Distance (Earth's Surface)
The Earth is a sphere, so straight-line distance through the Earth
isn't useful for navigation. Use the Haversine formula:
a = sin^2(Δlat/2) + cos(lat1) x cos(lat2) x sin^2(Δlon/2)
c = 2 x atan2(sqrt(a), sqrt(1-a))
d = R x c
(R = 6,371 km, Earth's mean radius)
Example: London (51.51°N, -0.13°E) to New York (40.71°N, -74.01°E)
Δlat = (40.71 - 51.51) x π/180 = -0.1886 rad
Δlon = (-74.01 - (-0.13)) x π/180 = -1.289 rad
a = sin^2(-0.0943) + cos(0.8993) x cos(0.7104) x sin^2(-0.6445)
= 0.00888 + 0.6272 x 0.7536 x 0.36265
= 0.00888 + 0.17148 = 0.18036
c = 2 x atan2(0.4247, 0.8955) = 0.8745 rad
d = 6,371 x 0.8745 = 5,571 km