Representing Solids

Learning Objectives

• Identify isometric, orthographic, cross-sectional views of solids.
• Draw isometric, orthographic, cross-sectional views of solids.
• Identify, draw, and construct nets for solids.

Introduction

The best way to represent a three-dimensional figure is to use a solid model. Unfortunately, models are sometimes not available. There are four primary ways to represent solids in two dimensions on paper. These are:

• An isometric (or perspective) view.
• An orthographic or blow-up view.
• A cross-sectional view.
• A net.

Isometric View

The typical three-dimensional view of a solid is the isometric view. Strictly speaking, an isometric view of a solid does not include perspective. Perspective is the illusion used by artists to make things in the distance look smaller than things nearby by using a vanishing point where parallel lines converge.

The figures below show the difference between an isometric and perspective view of a solid.

As you can see, the perspective view looks more “real” to the eye, but in geometry, isometric representations are useful for measuring and comparing distances.

The isometric view is often shown in a transparent “see-through” form.

Color and shading can also be added to help the eye visualize the solid.

Example 1

Show isometric views of a prism with an equilateral triangle for its base.

Example 2

Show a see-through isometric view of a prism with a hexagon for a base.

Orthographic View

An orthographic projection is a blow-up view of a solid that shows a flat representation of each of the figure’s sides. A good way to see how an orthographic projection works is to construct one. The (non-convex) polyhedron shown has a different projection on every side.

To show the figure in an orthographic view, place it in an imaginary box.

Now project out to each of the walls of the box. Three of the views are shown below.

A more complete orthographic blow-up shows the image of the side on each of the six walls of the box.

The same image looks like this in fold out view.

Example 3

Show an orthographic view of the figure.

First, place the figure in a box.

Now project each of the sides of the figure out to the walls of the box. Three projections are shown.

You can use this image to make a fold-out representation of the same figure.

Cross Section View

Imagine slicing a three-dimensional figure into a series of thin slices. Each slice shows a cross-section view.

The cross section you get depends on the angle at which you slice the figure.

Example 4

What kind of cross section will result from cutting the figure at the angle shown?

Example 5

What kind of cross section will result from cutting the figure at the angle shown?

Example 6

What kind of cross section will result from cutting the figure at the angle shown?

Nets

One final way to represent a solid is to use a net. If you cut out a net you can fold it into a model of a figure. Nets can also be used to analyze a single solid. Here is an example of a net for a cube.

There is more than one way to make a net for a single figure.

However, not all arrangements will create a cube.

Example 7

What kind of figure does the net create? Draw the figure.

The net creates a box-shaped rectangular prism as shown below.

Example 8

What kind of net can you draw to represent the figure shown? Draw the net.

A net for the prism is shown. Other nets are possible.

Multimedia Link The applet here animates how four solids are made from nets. There are two unique nets for the cube and two for the dodecahedron. Unfolding Polyhedra.

The following questions are for your own review. The answers below are for you to check your work and understanding.

Review Questions

1. Name four different ways to represent solids in two dimensions on paper.
2. Show an isometric view of a prism with a square base.

Given the following pyramid:

3. If the pyramid is cut with a plane parallel to the base, what is the cross section?
4. If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is the cross section?
5. If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the cross section?

Sketch the shape of the plane surface at the cut of this solid figure.

6. Cut
7. Cut
8. For this figure, what is the cross section?

   

Draw a net for each of the following:

9. 
10. 

Review Answers

1. Name four different ways to represent solids in two dimensions on paper.

Isometric, orthographic, cross sectional, net

2. Show an isometric view of a prism with a square base.

   

Given the following pyramid:

3. If the pyramid is cut with a plane parallel to the base, what is the cross section? square
4. If the pyramid is cut with a plane passing through the top vertex and perpendicular to the base, what is the cross section? triangle
5. If the pyramid is cut with a plane perpendicular to the base but not through the top vertex, what is the cross section? trapezoid

Sketch the shape of the plane surface at the cut of this solid figure.

6. 
7. 
8. pentagon
9. 
10.