How Do You Find the Distance Between Two Points in 3D Space?

You find the distance between two points in 3D space by calculating the differences in their $x$-, $y$-, and $z$-coordinates, squaring them, adding them, and taking the square root.

The Distance Formula

For points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, the 3D distance formula is

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

Each coordinate difference represents displacement along one mutually perpendicular axis. Squaring makes every contribution non-negative, so a negative change in $z$ still increases the total distance. The final square root converts the sum of squared changes back into the original coordinate units.

This formula is an extension of Pythagoras' theorem. First, the horizontal distance in the (xy)-plane is

$$d_{xy}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$$

This horizontal distance and the vertical change $z_2-z_1$ form another right-angled triangle. Therefore,

$$AB^2=d_{xy}^2+(z_2-z_1)^2,$$

which produces the 3D distance formula. Equivalently, use Pythagoras' theorem in the (xy)-plane, then use it again with the $z$-change. This explains why there are three squared terms rather than a direct sum.

Worked Example

Find the distance between $A(2,-1,4)$ and $B(7,3,-2)$.

Calculate each coordinate difference:

$$\Delta x=7-2=5,\qquad \Delta y=3-(-1)=4,\qquad \Delta z=-2-4=-6.$$

Substitute into the formula:

$$AB=\sqrt{5^2+4^2+(-6)^2}=\sqrt{77}.$$

Using a GDC, $AB\approx 8.77$. The distance is therefore $8.77$ units, correct to three significant figures. The order of subtraction does not matter because each difference is squared. If the coordinates were measured in metres, the answer would be $8.77\text{ m}$.

Quick Checks

If one coordinate is unchanged, its squared difference is zero; the same formula still applies. If all three coordinates match, the distance is zero. Keep the exact form $\sqrt{77}$ unless a decimal is requested. Never round coordinate differences, and check that the answer is non-negative and has linear, not squared, units.

Exam Technique

This is SL 3.1: 3D space, volume, angles and midpoints, common to AI SL and HL. Show the substituted formula before giving the GDC value, enter the complete square-root expression, retain full precision until the end, and include units. A common misconception is adding the coordinate differences directly; distance requires their squares followed by a square root.