Distance Between Two Points in 3D Calculator

The 3D distance formula

The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in space is:

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

This is the Pythagorean theorem applied twice. First, project both points onto the $xy$-plane: the horizontal difference $\Delta x = x_2 - x_1$ and depth difference $\Delta y = y_2 - y_1$ form the legs of a right triangle whose hypotenuse is the planar distance $\sqrt{\Delta x^2 + \Delta y^2}$. Second, that planar distance and the vertical difference $\Delta z = z_2 - z_1$ form a second right triangle — its hypotenuse is the full 3D distance.

Step by step

To find the distance between two points in space:

1. Subtract the x-coordinates: $\Delta x = x_2 - x_1$.
2. Subtract the y-coordinates: $\Delta y = y_2 - y_1$.
3. Subtract the z-coordinates: $\Delta z = z_2 - z_1$.
4. Square all three differences and add them: $\Delta x^2 + \Delta y^2 + \Delta z^2$.
5. Take the square root of that sum — simplifying it to a smaller radical when possible, or rounding it to a decimal approximation.

Worked examples

Example 1 — irrational result, no perfect-square factor to pull out: for $(7,4,3)$ and $(17,6,2)$, $\Delta x = 10$, $\Delta y = 2$, $\Delta z = -1$, so $d = \sqrt{10^2+2^2+(-1)^2} = \sqrt{100+4+1} = \sqrt{105} \approx 10.2470$.

Example 2 — another irrational, square-free result: for $(5,6,2)$ and $(-7,11,-13)$, $\Delta x = -12$, $\Delta y = 5$, $\Delta z = -15$, so $d = \sqrt{(-12)^2+5^2+(-15)^2} = \sqrt{144+25+225} = \sqrt{394} \approx 19.8494$.

Example 3 — perfect-square sum, exact whole number: for $(0,0,0)$ and $(1,2,2)$, $\Delta x = 1$, $\Delta y = 2$, $\Delta z = 2$, so $d = \sqrt{1^2+2^2+2^2} = \sqrt{9} = 3$ exactly.

Example 4 — identical points: two points with the same $x$, $y$, and $z$ coordinates are zero distance apart — every difference is $0$, so $d = \sqrt{0} = 0$.

Exact form vs. decimal approximation

Because most coordinate pairs are not a whole number of units apart, the exact 3D distance is usually an irrational number — one that never terminates or repeats as a decimal. This calculator shows both forms: the exact simplified radical (e.g. $\sqrt{105}$), reduced the same way the Simplify Radicals calculator pulls perfect-square factors out of a radicand, and a rounded decimal approximation beneath it (e.g. $\approx 10.246951$) for whichever a problem calls for.

Related calculators

This calculator is the direct 3D extension of Distance Between Two Points, which finds the same kind of distance on a flat plane (drop the $z$-coordinate and the two formulas match exactly). It also shares its formula with the Rectangular Prism calculator's space diagonal — the distance between two opposite corners of a box is $\sqrt{l^2+w^2+h^2}$, the same expression with $l$, $w$, $h$ in place of $|\Delta x|$, $|\Delta y|$, $|\Delta z|$. The underlying right-triangle relationship comes from the Pythagorean Theorem calculator, applied twice here instead of once. The same two points also determine a Midpoint — the point exactly halfway between them in space.