6. Geometry & Measurement: Solids

Lesson

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.

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:

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.

Choose the net that folds to give the shape below:

- ABCDABCD

Choose the shape that has the following net:

- ABCDABCD

Three-dimensional objects can be represented with the front view (which is sometimes called the front elevation), the side view (which is sometimes called the side elevation) and the top view (which is sometimes call a plan view) 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:

(a) What is the front view?

(b) What is the top (or plan) view?

(c) What is the side view?

**(a) Think:** We can color the sides of the cubes that are facing the front to visualize this

**Do:** Color the front faces of the image like this:

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

Front view |

**(b) Think: ** We can do the same from above by coloring the top sides of the image.

**Do: ** Coloring the top sides of the image we get this.

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

Plan view |

**(c) Think:** When thinking about the side view, we can again use the same trick.

**Do: **

** Reflect:** 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 |

Consider this solid formed from cubes. Note that the "top” view of a figure is sometimes called a "plan" view.

Which of the following diagrams represents the plan view?

ABCABCWhich of the following diagrams represents the side view?

ABCABCWhich of the following diagrams represents the front view?

ABCDABCD

Construct a three-dimensional model, given the top or bottom, side, and front views