Distance Formula Calculator
Find the straight-line distance between two points on a coordinate plane, or in 3D space, with every step of the distance formula shown.
- Δx = 6, Δy = 8
- Δx² + Δy² = 36 + 64 = 100
- d = √100 = 10
- Horizontal Δx
- 6
- Vertical Δy
- 8
- Midpoint
- (1, 5)
- Slope
- 1.3333
The distance formula is Pythagoras on a grid
Draw a right triangle with the two points at the ends of the hypotenuse. The legs are the horizontal gap Δx and the vertical gap Δy, so the distance is √(Δx² + Δy²). In 3D, add a third term: √(Δx² + Δy² + Δz²). The dashed step in the diagram is that right triangle.
Example: from (−2, 1) to (4, 9), Δx = 6 and Δy = 8, so d = √(36 + 64) = 10.
This is straight-line ("as the crow flies") distance on a flat plane. Travel distance between cities follows roads, and distances over the Earth's surface use the great-circle (haversine) formula instead.
Questions people ask
What is the distance formula?
d = √((x₂ − x₁)² + (y₂ − y₁)²). It comes straight from the Pythagorean theorem.
Can distance be negative?
No. The differences are squared before adding, so the result is always zero or positive.
How do I find the distance between two points in 3D?
Add the z term: d = √(Δx² + Δy² + Δz²). Switch the calculator to 3D.
Is this a driving distance calculator?
No. It measures straight-line distance between coordinates on a flat grid, as used in algebra and geometry.
Stuck on a word problem?
Paste a homework question or describe your project. The AI tutor names the formula and works it through step by step. Always check the result with the calculator above.