Topics
10. Measurement
topic badge
Middle Years

10.07 Visualising solids

Lesson
Worksheet
Practice
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.

 

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

Choose the net that folds to give the shape below:

A rectangular prism shaded in blue. Three faces are visible from view, each shaded in varying shades of blue.

  1. A dark blue net consisting of one horizontal rectangle, one pair of equilateral triangles, and one pair of isosceles triangles. An inverted equilateral triangle is at the center. A horizontal rectangle is attached above the central triangle. An isosceles triangle is attached to the right side of the central triangle. Another isosceles triangle is attached to the upper right side of the first isosceles triangle. The two isosceles triangles are connected at their equal sides. An equilateral triangle is attached to the bottom right side of the first isosceles triangle.

    A

    A dark blue net consisting of three pairs of identical rectangles. The largest rectangles are attached to each other in the center, one below the other. A horizontal rectangle is attached above the upper central rectangle. Another horizontal rectangle is attached below the lower central rectangle. A vertical rectangle is attached to the right side of the central rectangle, with a second vertical rectangle attached to the right side of the first vertical rectangle.

    B

    A dark blue net consisting of one horizontal rectangle, a pair of identical equilateral triangles, and a pair of identical isosceles triangles. The rectangle is at the center. An equilateral triangle is attached to the top, and another is attached to the bottom. An isosceles triangle is attached to the left side of the upper equilateral triangle. An isosceles triangle is attached to the right side of the lower equilateral triangle.

    C

    A dark blue net consisting of three pairs of identical rectangles. The central shape is one of the largest rectangles. A pair of identical vertical rectangles is attached to the left and right sides of the central rectangle. A pair of identical horizontal rectangles is attached above and below the central rectangle. The other largest rectangle is attached below the lower horizontal rectangle.

    D

Question 2

Choose the shape that has the following net:

The net of a square pyramid featuring one square that forms the base connected to 4 distinct triangles that when folded up form a square pyramid

  1. a rectangular pyramid

    A

    a square pyramid

    B

    a rectangular prism

    C

    a square prism

    D

 

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

Consider this solid formed from cubes:

A solid shaded in green is formed from six cubes. Three arrows point in different directions, indicating the top view, side view, and front view. The top arrow points to the top face of the solid and is labeled "Top View". The left arrow points to the left face of the solid and is labeled "Side View". The right arrow points to the front face of the solid and is labeled "Front View". The arrangement consists of five cubes forming the base and one cube positioned on top. On the base, three cubes are arranged in a row, with one cube placed in front of the left cube and another cube behind the right cube. The sixth cube is positioned above the cube behind the right cube.

  1. Which of the following diagrams represents the plan view?

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the right square.

    A

    A view of a solid consists of five green squares. Three squares are aligned vertically. The fourth square is on the left of the top square. The fifth square is on the right of the bottom square.

    B

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the left square.

    C

  2. Which of the following diagrams represents the side view?

    A view of a solid consists of five green squares. Three squares are aligned vertically. The fourth square is on the left of the top square. The fifth square is on the right of the bottom square.

    A

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the left square.

    B

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the right square.

    C

  3. Which of the following diagrams represents the front view?

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the middle square.

    A

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the left square.

    B

    A view of a solid consists of five green squares. Three squares are aligned vertically. The fourth square is on the left of the top square. The fifth square is on the right of the bottom square.

    C

    A view of a solid consists of four green squares. Three squares are at the bottom. The fourth square is above the right square.

    D
 

What is Mathspace

About Mathspace