10.07 Visualising solids

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.

There are many ways to unfold a net from a solid, and in this chapter we will investigate nets of prisms and pyramids.

Here are some prisms:

Prisms have rectangular sides, and the shape on the top and the base is the same. The name of this shape gives the prism its name. Any cross-section taken parallel to the base is always the same.

Here are some pyramids:

Pyramids have triangular sides, and the shape on the base gives the pyramid its name. Any cross-section taken parallel to the base is always the same shape, but is smaller in size than the base.

Examples

Example 1

Choose the net that folds to give the shape below:

A

B

C

D

Worked Solution

Create a strategy

Choose the net that has the same number of faces as the solid and can be folded up along its edges to make the solid.

Apply the idea

The answer is option D.

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Example 2

Choose the shape that has the following net:

A

B

C

D

Worked Solution

Apply the idea

Since there is only one square, it must be a square pyramid.

The answer is option B.

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Three-dimensional objects can be represented with the side elevation, front elevation, and top elevation (called plan) clearly indicated on a two-dimensional surface. We can then ask about the view from each of these elevations.

Consider this solid formed from cubes:

What is the front view?

We can colour the sides of the cubes that are facing the front to make an image like this.

We can now piece together the front view by joining the highlighted faces together.

We can do the same from above to get the plan view.

Looking up and over from the side, we can tell that this is the plan view.

When thinking about the side view, we can again use the same trick.

However, this time there is a highlighted face that would be hidden from the side that we don't include.

Once we have identified this hidden side, we can draw the side view properly.

Examples

Example 3

Consider this solid formed from cubes:

a

Which of the following diagrams represents the plan view?

A

B

C

Worked Solution

Create a strategy

Try imagining the tops of each cube as a different colour:

Apply the idea

Joining the highlighted faces together, the diagram which represents the plan view is given by:

The correct answer is Option B.

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b

Which of the following diagrams represents the side view?

A

B

C

Worked Solution

Create a strategy

Try imagining the sides of each cube as a different colour:

Apply the idea

Joining the highlighted sides of each cube, the diagram which represents the side view is given by:

The correct answer is Option B.

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c

Which of the following diagrams represents the front view?

A

B

C

D

Worked Solution

Create a strategy

Try imagining the fronts of each cube as a different colour:

Apply the idea

Joining the highlighted fronts of each cube, the diagram which represents the front view is given by:

The correct answer is Option D.

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