Three-dimensional objects are represented on two-dimensional surfaces all the time. Screens, whiteboards, paper, and other flat surfaces can create the illusion of depth when displaying a picture of something.
There are a few tricks we can use to think about three-dimensional objects represented on a flat surface.
Nets
We can never see every part of a three-dimensional object at once - there is always part of it that is behind the view we are looking at. To better think about a solid object we sometimes represent it with its net. Each face of the solid is laid flat on the same surface, breaking it along the edges and folding it out. This way we can think about folding it back up along its edges to recover the original shape.
Here is a triangular prism. Move the slider to see its net unfold:
Practice questions
Question 1
Choose the net that folds to give the shape below:
A
B
C
D
Question 2
Choose the shape that has the following net:
A
BC
D
Different views of prisms
When looking at a prism, we can look at the prism from a 3D view or from one of the 2D views.
A 3D view of the prism shows us what the whole solid looks like from an angle. From this perspective, we can see the faces that will be visible when looked at directly from one of the 2D views.
The 2D views show what the solid looks like when viewed directly from the front, side or top.
In the 3D view of this rectangular prism we can see the sides that will be visible in the 2D views. In each 2D view we look at the rectangular prism directly from either the front, side or top.
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 |
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| 3D view | Front View | Side View | Plan View |
As we can see from the images, the front, side and top views are all 2D shapes that match the faces of the prism visible from each view when looked at directly.
The plan view is another name for the top view and is used in architecture when referring to the plan of a building, which is how the building looks when viewed from directly above.
An elevation can be used to describe the front, back or side views. Elevations are used in architecture to show how the building looks when viewed from the front, back or sides.
Viewing sloped and curved sides
Since the 2D views of a solid only show what the solid looks like directly from one angle, these views cannot show depth like how a 3D view can.
For example, if we look at this hexagonal prism from the front, we will be able to see these three faces.
The faces visible from the front view of the hexagonal prism
However, two of these faces are sloped and will appear thinner in the front view than they actually are. As a result, the front view will look like this:
The front view of the hexagonal prism
This is because the sloped sides aren't as wide when viewed directly from the front. As is shown in the diagram below, the distance between the two ends of the sloped sides is closer together when viewed from the front because the 2D view doesn't show that one end is further away than the other.
Looking at sloped sides directly
For a similar reason, the side view of a cylinder will look like a rectangle:
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| 3D view | Front View |
Curved sides in 3D will always look flat in a 2D view.
Looking at curved sides directly
Viewing other sides
A 2D view might also show a side that we can't see from the 3D view.
When looking at a triangular prism from the 3D view, we notice that we can only see two of the five faces. The bottom and back faces of the prism won't show up on any of our 2D views but the last hidden face will be visible from the top view.
The 3D view of a triangular prism
Looking at these two sloped faces from directly above, like so:
Looking at the visible and hidden faces directly
We find that the top view of this triangular prism looks like this:
The top view of the triangular prism
The fact that we can see two faces from the top view is shown by the line dividing the view into two rectangular faces.
Practice questions
Question 3
Consider the different views of this trapezoidal prism.
What is the front view?
A
B
C
DWhat is the side view?
ABCDWhat is the top view?
ABCD
Question 4
Consider the different views of this composite solid.
What is the front view?
A
B
C
DWhat is the side view?
ABC
DWhat is the plan view?
ABC
D
Question 5
Match the front view to the correct solid:
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| Front View |
- A
B
C
D