Q1. Find the distance between (1, 2, 3) and (4, 6, 8).
d = √[(4−1)² + (6−2)² + (8−3)²]
d = √[9 + 16 + 25] = √50
d = 5√2 units
Introduction to Three Dimensional Geometry — Class 11
Add a third axis and everything from 2D geometry extends naturally. Compute the distance between any two points in space.
Three coordinate axesx, y and z axes, mutually perpendicular, meeting at the origin O(0,0,0). Space is divided into 8 octants.
Distance formula (3D)Between (x₁,y₁,z₁) and (x₂,y₂,z₂): d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
Section formula (3D)The point dividing the join in ratio m:n has coordinates ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)).
3D distance calculator Interactive
Enter two points in space to find the distance between them.
Distance—
This is exactly the 2D distance formula with one extra (z₂−z₁)² term added under the root.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Distance in 3D | d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²] |
| Midpoint in 3D | ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) |
Common mistakes & exam wins
- The 3D distance formula is just Pythagoras applied twice — add one more squared-difference term to the 2D formula.
- Space is divided into 8 octants (compare to 4 quadrants in 2D).
- A point on the x-axis has both y = 0 and z = 0.
Frequently asked questions
What is the distance formula in three dimensions?
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²], extending the 2D formula with a z-term.
How many octants does 3D space have?
Eight, formed by the three coordinate planes.
What are the coordinates of a point on the y-axis?
(0, y, 0) — both the x and z coordinates are zero.