Distance Formula Calculator

The formulas

2D distance between (x1, y1) and (x2, y2):

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

3D distance between (x1, y1, z1) and (x2, y2, z2):

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Both are direct consequences of the Pythagorean theorem: the squared distance equals the sum of squared differences along each axis.

Worked example

From (3, 4) to (7, 1):

• dx = 7 - 3 = 4
• dy = 1 - 4 = -3
• d = sqrt(16 + 9) = sqrt(25) = 5

In 3D from (1, 2, 3) to (4, 6, 3):

• dx = 3, dy = 4, dz = 0
• d = sqrt(9 + 16 + 0) = sqrt(25) = 5

When to use which

• 2D covers screen coordinates, map distances at small scale (where Earth curvature is negligible), and most homework problems.
• 3D is for simulations, game engines, molecular distances and CAD measurements.
• Great-circle distance on the surface of Earth requires the haversine formula, not the 3D Pythagorean one, because the path follows a curve. Use a dedicated geographic distance tool.

Common mistakes

• Forgetting the square root. Squaring once is not enough; you need to take the root at the end.
• Order of subtraction. (x2 - x1) vs (x1 - x2) does not matter because the result is squared, but keep it consistent with the textbook you are checking against.
• Mixing units. All coordinates must be in the same unit; mixing feet and meters silently gives a meaningless number.
• Negative coordinates. Double-negatives become positives inside the square, so watch the signs on points in quadrants II, III or IV.