10.07 Visualising solids

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:

Created with Geogebra

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
Triangular	Square	Rectangular	Pentagonal	Hexagonal	Octagonal

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
Triangular	Square	Rectangular	Pentagonal	Hexagonal	Octagonal

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

Practice questions

Question 1

Question 2

 

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.

Worked example

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: 

Front view

We can do the same from above:

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

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: 

Side view

Practice question

Question 3