In 3D coordinates, every point in space is described by three numbers written as (x, y, z). The x-axis goes left–right, the y-axis goes forwards–backwards, and the z-axis goes up–down. Once you understand how 2D coordinates work in a flat plane, extending to three dimensions follows naturally.
How do 2D and 3D coordinates compare?
In 2D you need two numbers to locate a point on a flat plane. In 3D you need a third number to say how high the point is above (or below) that plane.
| Feature | 2D coordinates | 3D coordinates |
|---|---|---|
| Format | (x, y) | (x, y, z) |
| Number of axes | 2 (horizontal, vertical) | 3 (left-right, front-back, up-down) |
| Origin | (0, 0) | (0, 0, 0) |
| How many numbers to locate a point | 2 | 3 |
| Used for | Flat maps, graphs | Space, solid geometry, 3D shapes |
The origin in 3D is the point where all three axes meet: (0, 0, 0). All three axes are perpendicular (at 90°) to each other.
What does each coordinate mean?
For a point written as (x, y, z):
- x tells you how far to move along the x-axis (positive = right, negative = left).
- y tells you how far to move along the y-axis (positive = forwards, negative = backwards).
- z tells you how far to move along the z-axis (positive = up, negative = down).
The order is always x, then y, then z — just as in 2D you always write x before y.
Example: The point (3, 2, 5) is 3 units to the right, 2 units forwards and 5 units up from the origin.
How do you plot a point in 3D?
To plot (4, 3, 2) on a 3D grid:
- Start at the origin (0, 0, 0).
- Move 4 units in the positive x-direction (along the x-axis, going right).
- From there, move 3 units in the positive y-direction (forwards).
- From there, move 2 units in the positive z-direction (upwards).
- Mark the point.
On a 3D diagram on paper, the axes are usually drawn at angles to give a sense of depth. The x-axis typically goes to the right, the y-axis goes diagonally into the page, and the z-axis goes vertically upwards.
How do you find the coordinates of vertices of a 3D shape?
Worked example 1: A cuboid has one corner at the origin and the opposite corner at (5, 4, 3). Write the coordinates of all eight vertices.
The cuboid is 5 units wide (x), 4 units deep (y) and 3 units tall (z). The eight corners are every combination of 0 or the maximum value for each axis:
| Vertex | x | y | z | Coordinate |
|---|---|---|---|---|
| A | 0 | 0 | 0 | (0, 0, 0) |
| B | 5 | 0 | 0 | (5, 0, 0) |
| C | 5 | 4 | 0 | (5, 4, 0) |
| D | 0 | 4 | 0 | (0, 4, 0) |
| E | 0 | 0 | 3 | (0, 0, 3) |
| F | 5 | 0 | 3 | (5, 0, 3) |
| G | 5 | 4 | 3 | (5, 4, 3) |
| H | 0 | 4 | 3 | (0, 4, 3) |
The bottom face (z = 0) uses vertices A, B, C, D. The top face (z = 3) uses E, F, G, H.
How do you find the midpoint of a line in 3D?
The midpoint between two points in 3D uses exactly the same rule as in 2D: average each coordinate separately.
Midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2)
Worked example 2: Find the midpoint of A(2, 4, 6) and B(8, 2, 0).
- x: (2+8)/2 = 5
- y: (4+2)/2 = 3
- z: (6+0)/2 = 3
Midpoint = (5, 3, 3)
How do you find the distance between two points in 3D?
At KS3 you may be introduced to the 3D version of Pythagoras' theorem:
Distance = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)
Worked example 3: Find the distance between P(1, 2, 3) and Q(4, 6, 3).
- Differences: x: 4−1=3, y: 6−2=4, z: 3−3=0
- Sum of squares: 3² + 4² + 0² = 9 + 16 + 0 = 25
- Distance = √25 = 5 units
Notice that when z is the same for both points (z difference = 0), the formula reduces to the 2D version — a 3² + 4² + 0² is just 3² + 4² = 25, confirming that the 3D formula is a natural extension of the 2D one.
Frequently asked questions
Why is the z-axis needed if a 2D grid already covers flat surfaces?
A 2D grid can describe any point on a flat surface (such as a map or a piece of paper). To describe a point that is above or below that surface — like a position in a room or the vertex of a 3D solid — you need a third number, the z-coordinate, to record the height.
Does it matter which axis goes which direction?
The convention used in most GCSE and KS3 resources is: x goes right, y goes forwards (into the page in a 3D diagram), z goes up. Some science and engineering texts swap y and z. Unless your teacher specifies otherwise, use the convention in the diagram given.
What is the coordinate of a point on the x-axis?
Any point on the x-axis has y = 0 and z = 0. For example, the point 7 units along the x-axis is (7, 0, 0). Similarly, a point on the y-axis has x = 0 and z = 0, and a point on the z-axis has x = 0 and y = 0.
How does 3D coordinate geometry link to vectors?
A 3D coordinate (x, y, z) can also be written as a position vector from the origin. You will study this connection at GCSE and A-level, where vectors in three dimensions are used to find distances, angles and directions in space. The 3D Pythagoras formula you saw above is closely related to the magnitude of a 3D vector.
For KS3 geometry and coordinate skills with Professor Pi — visit aitutors.me.