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10.07 Visualising solids

Lesson

Introduction

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.

Exploration

Here is a triangular prism. Move the slider to see its net unfold:

Loading interactive...

The net of a triangular prism is made up of three rectangles and two triangles.

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:

Table of different kinds of prisms containing triangular, square, rectangular, pentagonal, hexagonal, and octagonal 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:

Table of kinds of pyramids containing triangular, square, rectangular, pentagonal, hexagonal, and octagonal 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:

Image of rectangular prism.
A
This image shows a net of a solid. Ask your teacher for more information.
B
This image shows a net of a solid. Ask your teacher for more information.
C
This image shows a net of a solid. Ask your teacher for more information.
D
This image shows a net of a solid. Ask your teacher for more information.
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.

This image shows a net of a solid. Ask your teacher for more information.

Example 2

Choose the shape that has the following net:

This image shows a net of a solid. Ask your teacher for more information.
A
Image of rectangular pyramid.
B
Image of square pyramid.
C
Image of rectangular prism.
D
Image of a cube.
Worked Solution
Apply the idea

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

The answer is option B.

This image shows a net of a solid. Ask your teacher for more information.
Idea summary

Prisms have rectangular sides, and the shape on the top and the base is the same. Any cross-section taken parallel to the base is always the same.

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

Front, side, and plan view

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:

3 dimensional object made of blocks with  side, plan, and front elevation labelled. Ask your teacher for more information.

What is the front view?

3 dimensional object made of blocks with  front elevation highlighted. Ask your teacher for more information.

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

A front view of the three-dimensional object. Ask your teacher for more information.

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

3 dimensional object made of blocks with  plan elevation highlighted. Ask your teacher for more information.

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

A plan view of the three-dimensional object. Ask your teacher for more information.

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

3 dimensional object made of blocks with  side elevation highlighted. Ask your teacher for more information.

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.

A side view of the three-dimensional object. Ask your teacher for more information.

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

Examples

Example 3

Consider this solid formed from cubes:

3 dimensional object with side, plan, and front elevation labelled. Ask your teacher for more information.
a

Which of the following diagrams represents the plan view?

A
4 squares in a backwards L shape. Ask your teacher for more information.
B
5 squares making a shape. Ask your teacher for more information.
C
4 squares in an L shape. Ask your teacher for more information.
Worked Solution
Create a strategy

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

A solid formed from cubes highlighting the plan view. Ask your teacher for more information.
Apply the idea

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

5 squares making a shape. Ask your teacher for more information.

The correct answer is Option B.

b

Which of the following diagrams represents the side view?

A
5 squares making a shape. Ask your teacher for more information.
B
4 squares in an L shape. Ask your teacher for more information.
C
4 squares in a backwards L shape. Ask your teacher for more information.
Worked Solution
Create a strategy

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

A solid formed from cubes highlighting the side view. Ask your teacher for more information.
Apply the idea

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

A diagram representing the side view of the solid formed from cubes. Ask your teacher for more information.

The correct answer is Option B.

c

Which of the following diagrams represents the front view?

A
4 squares making a shape. Ask your teacher for more information.
B
4 squares in an L shape. Ask your teacher for more information.
C
5 squares making a shape. Ask your teacher for more information.
D
4 squares in backwards L shape. Ask your teacher for more information.
Worked Solution
Create a strategy

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

A solid formed from cubes highlighting the front view. Ask your teacher for more information.
Apply the idea

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

A diagram representing the  front  view of the solid formed from cubes. Ask your teacher for more information.

The correct answer is Option D.

Idea summary

Three-dimensional objects can be represented with the side elevation, front elevation, and top elevation (called plan):

3 dimensional object made of blocks with  side, plan, and front elevation labelled. Ask your teacher for more information.

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