Library / Foundations: Maths / FMA-36
Foundations: Maths
Distance between two points
Pythagoras applied to the coordinate differences between two points.
- Ref
- FMA-36
- Rev
- v1
- Field
- Foundations: Maths
- Result
- d [m]
(x₁, y₁) and (x₂, y₂) are the two points in the same grid, with both coordinates in the same unit. The distance is the hypotenuse of a right-angled triangle whose legs are the east–west and north–south differences. The bearing is measured clockwise from grid north.
Pythagoras applied to the coordinate differences between two points.
Calculator
- East–west difference Δx3 m
- North–south difference Δy4 m
- Bearing from grid north36.87 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Moving both points by the same amount changes nothing, because only the differences enter.
- Cutting the corner saves distance: the default route along the two legs is 7 m, but the straight line is 5 m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| y1 y of first point | −1,000 to 1,000 m | 5 | 1,006 | 1,001 | +40 m | +622 % |
| x1 x of first point | −1,000 to 1,000 m | 5 | 1,004 | 999 | +40 m | +644 % |
| y2 y of second point | −1,000 to 1,000 m | 3.606 | 1,002 | 998.4 | +40 m | +782 % |
| x2 x of second point | −1,000 to 1,000 m | 4.123 | 1,001 | 996.9 | +40 m | +764 % |
Sweep of y of second point
| y of second point (m) | Distance (m) |
|---|---|
| −1,000 | 1,002 |
| −750 | 752 |
| −500 | 502 |
| −250 | 252 |
| 0 | 3.606 |
| 250 | 248 |
| 500 | 498 |
| 750 | 748 |
| 1,000 | 998 |
Sweep one variable
Variables and how each one contributes
Distance
What the equation gives you. Everything else feeds it.
x of first point
East–west coordinate of the first point. Only the difference x₂ − x₁ matters. Moving the first point away from the second raises the distance; moving it towards the second lowers it.
y of first point
North–south coordinate of the first point. Only the difference y₂ − y₁ matters. Moving the first point away from the second raises the distance; moving it towards the second lowers it.
x of second point
East–west coordinate of the second point. Only the difference x₂ − x₁ matters. Moving the second point away from the first raises the distance.
y of second point
North–south coordinate of the second point. Only the difference y₂ − y₁ matters. Moving the second point away from the first raises the distance.
Worked example with the values above
- Differences: Δx = 4 − 1 = 3 m, Δy = 6 − 2 = 4 m.
- Squares: 9 + 16 = 25.
- d = √25 = 5 m.
Assumptions and limits
- Both points are on one flat grid with the same scale on both axes.
- Coordinates are in the same unit; over long distances the curvature of the Earth makes a grid distance differ from the true one.
Common mistakes
- Adding the two differences instead of squaring, adding and rooting.
- Reading eastings and northings in the wrong order for one of the points.
Statistics and uncertainty
If each coordinate of each point carries an independent error σ, the error in the straight-line distance is about √2 σ along the line joining them, whatever the distance. A grid reference good to ±10 m therefore gives a separation good to about ±14 m.
Where it is used
- Distance between two grid references.
- Range from an observer to a target in map coordinates.
Maths you need
- Pythagoras: a² + b² = c²
- Coordinates: x and y positions on a grid