Comparing diagrams, views and digital representations | Year 8

01 / LEARN

Understand the idea.

Define the viewing direction

A flat view hides one coordinate, so first state the viewing convention. Here, front means looking along $y$ and showing $(x, z)$ ; side means looking along $x$ and showing $(y, z)$ ; top means looking along $z$ and showing $(x, y)$ .

Keep the names of the retained axes attached to each pair. For $P (3, 2, 4)$ , the front view keeps across and height, the side keeps depth and height, and the top keeps across and depth. Dropping an entry does not reorder the others.

Labelled x, y and z axes have numbered one-unit steps. A dashed gold route reaches P by following the x direction, then the y direction, then the z direction. The perspective projection does not preserve spatial lengths.

Views of P(3,2,4) under the stated convention
View	Shown coordinates	Hidden coordinate
Front	(3,4) on x,z	y=2
Side	(2,4) on y,z	x=3
Top	(3,2) on x,y	z=4

Compare representations rather than appearances

Different 3D points can overlap in one view because their hidden coordinates differ. Points $(2, 1, 3)$ and $(2, 5, 3)$ both appear at $(2, 3)$ in the front view. Another view can reveal their different depths.

To combine views, assign every shown number to its named axis and check shared coordinates agree. A rotated screen view can look different while preserving the same stored triple. Coordinates change only if the point or its reference coordinate system changes.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three views of one point

Find the front, side and top views of P(3,2,4), using front=(x,z), side=(y,z) and top=(x,y).

Worked solution

1
Keep x and z for the front view.
$(x, z) = (3, 4)$
Looking along y hides the depth coordinate 2.
2
Keep y and z for the side view.
$(y, z) = (2, 4)$
Looking along x hides the across coordinate 3.
3
Keep x and y for the top view.
$(x, y) = (3, 2)$
Looking along z hides the height coordinate 4, giving front $(3, 4)$ , side $(2, 4)$ and top $(3, 2)$ .

Answer

Front (3,4); side (2,4); top (3,2).

EXAMPLE 02

Combine two views

A point has front view (4,2) on x,z and top view (4,3) on x,y. Find its triple.

Worked solution

1
Read the coordinates supplied by the front view.
$(x, z) = (4, 2) ⟹ x = 4, z = 2$
The front pair's second entry is height, not depth.
2
Read the coordinates supplied by the top view.
$(x, y) = (4, 3) ⟹ x = 4, y = 3$
The top view supplies the missing depth and agrees with the front view's x-coordinate.
3
Assemble the coordinates in their fixed order.
$(x, y, z) = (4, 3, 2)$
The combined views determine all three entries of the point.

Answer

(4,3,2).

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the top view of (2,5,3) on x,y?

Show answer & reasoning

(2,5)

The top view retains $x$ and $y$ while hiding $z$ . Therefore $(2, 5, 3)$ appears at $(2, 5)$ on the top-view axes.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find the front view of $(1, 4, 2)$ , where front shows $x, z$ .
2 marks
Find the side view of $(3, 2, 5)$ , where side shows $y, z$ .
2 marks
Find the top view of $(4, 1, 3)$ , where top shows $x, y$ .
2 marks
Front view on $x, z$ is $(2, 4)$ ; top view on $x, y$ is $(2, 3)$ . Find the triple.
2 marks
Do $A (2, 1, 3)$ and $B (2, 5, 3)$ overlap in a front view on $x, z$ ? Explain.
2 marks
A digital model is rotated on screen. Its coordinate axes rotate with it. Must a stored point’s coordinate triple change? Explain.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the front view of $(1, 4, 2)$ , where front shows $x, z$ .
2 marks
Retain the first and third coordinates.
$(1, 2)$
Find the side view of $(3, 2, 5)$ , where side shows $y, z$ .
2 marks
Retain the second and third coordinates.
$(2, 5)$
Find the top view of $(4, 1, 3)$ , where top shows $x, y$ .
2 marks
Retain the first and second coordinates.
$(4, 1)$
Front view on $x, z$ is $(2, 4)$ ; top view on $x, y$ is $(2, 3)$ . Find the triple.
2 marks
Front gives $x = 2$ , $z = 4$ ; top gives $y = 3$ .
$(2, 3, 4)$
Do $A (2, 1, 3)$ and $B (2, 5, 3)$ overlap in a front view on $x, z$ ? Explain.
2 marks
The view hides $y$ .
Yes; both show $(2, 3)$ , though their $y$ -coordinates differ.
A digital model is rotated on screen. Its coordinate axes rotate with it. Must a stored point’s coordinate triple change? Explain.
2 marks
Distinguish the screen view from a change of coordinate system.
No. Rotating the view alone does not change coordinates relative to the same axes.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Coordinate transformations review; identifying positions using three coordinates; interpreting 3D axes; describing routes and relative positions; comparing diagrams, views and digital representations. This is spatial location, not senior vector algebra.

Official curriculum source ↗

In this lesson

Understand the idea.

Define the viewing direction

Compare representations rather than appearances

See the method in action.

Three views of one point

Keep x and z for the front view.

Keep y and z for the side view.

Keep x and y for the top view.

Combine two views

Read the coordinates supplied by the front view.

Read the coordinates supplied by the top view.

Assemble the coordinates in their fixed order.

Try it yourself.

Check your understanding

Keep practising