3D coordinates

Video lesson2 worked examples

Theory

You must be able to determine the x, y, and z coordinates of a specific point from a diagram representing a 3-dimensional object (like a cuboid or pyramid).

The Golden Rule: read each coordinate as how far the point is along the $x$ , then $y$ , then $z$ axis from the origin. For a midpoint or a centre, take the average of the coordinates of the two ends (or of the corners).

⚠️ Common Examiner Traps

Order matters: always give coordinates as $(x, y, z)$ in that order.
Points on the floor: a point on the base has $z = 0$ ; a point directly above another shares its $x$ and $y$ .
Midpoints: the midpoint of an edge is the average of its two end coordinates — halve each difference, don't guess.
Centre of a base: the centre is the average of the base corners, which usually gives half-values.

Worked examples

Example 1

Identifying 3D Vertices

A rectangular room is modelled on a 3D coordinate system. The origin (0, 0, 0) represents the bottom-left corner of the floor. The x-axis represents the length (8 m), the y-axis represents the width (6 m), and the z-axis represents the height (3 m). State the coordinates of the ceiling corner diagonally opposite the origin.

Step 1: Travel along the x-axis to the end of the length: x = 8.

Step 2: Travel along the y-axis to the end of the width: y = 6.

Step 3: Travel up the z-axis to the ceiling: z = 3.

Answer: (8, 6, 3).

Example 2

Coordinates of a Midpoint

A cuboid sits with one corner at the origin. It measures 8 units along the $x$ -axis, 5 units along the $y$ -axis and 6 units along the $z$ -axis. Find the coordinates of the point M at the centre of the top face.

Step 1: The top face is at full height, so $z = 6$ .

Step 2: The centre is halfway across the face in both horizontal directions — average each: $x = \tfrac{8}{2} = 4$ and $y = \tfrac{5}{2} = 2.5$ .

Answer: M is $(4,\ 2.5,\ 6)$ .

Ready to practise?

12 3D Coordinates questions with answers and video solutions