Exploring Solid Shapes

Introduction

In the last lesson, we only used 6 square polydron pieces. Today we will use other ones to build the nets of different shapes.

Make a cube
- How many faces does it have?
- How many edges does it have?
- How many vertices does it have?
- How many faces meet at each vertex?
Let's now work with just the equilateral triangles pieces. Can you make a tetrahedron? (click on the hyperlink if you are not sure what it is.)
- How many faces does it have?
- How many edges does it have?
- How many vertices does it have?
- How many faces meet at each vertex?
- Record the information in the table on the right.
What other polyhedra can be made using just triangles?
- Make them, take pictures/ videos of them so you don't forget.
- Record the information in the table on the right.
- Look at these deltahedrons (a polyhedron whose faces are all equilateral triangles) - did you get most of them?
Is it possible to make more than one shape with the same number of pieces in some cases?
- How many different ones can you make?
- How do you know if you have found them all?
What is the connection between the number of faces and the number of edges?
- Can you explain why this happens?
- Why is it not possible to make a shape with an odd number of faces?
What is the connection between the number of vertices and the number of faces?
- Hint: think about how many faces there are at each vertex.
Find the sum of the number of faces and the number of vertices. How does this sum compare with the number of edges?
- Does this rule work for all your shapes?
- Hint: look at Euler's Formula.
Looking only at the polyhedra made with triangles, what is different about the ones that use 4 and 8 triangles?
- The shapes with 4 triangles (tetrahedron) and 8 (octahedron) are quite special and are known as Platonic Solids. These are described as completely regular.
- Are there any more completely regular solids that use only triangles?
- If there are, how many triangles are there at each vertex?
- How can you prove that these are the only possible shapes using triangles?
Hopefully, by completing Q8 you will have now found all the platonic solids made of equilateral triangles. The cube is another platonic solid with square sides. Can you explain why it is a platonic solid?
Get some pentagonal polydrons. Can you now find the last platonic solid with regular pentagonal sides?
- How many faces does it have?
- How many edges does it have?
- How many vertices does it have?
- How many faces meet at each vertex?
- Look at this from NCTM to explore the Platonic Solids.

Further Questions and Challenges

The angle defect at any vertex is the sum of the internal angles of the polygons at that vertex subtracted from 360.
- Calculate the angle defect at each vertex of the shapes and sum those values. What do you notice?
Start with a tetrahedron, remove one face and replace it with three faces of another tetrahedron.
- What does this do to the sum of the total number of faces?
- What does this do to the sum of the total number of edges?
- What does this do to the sum of the total number of vertices?
- Looking at the rule you derived in Q7, does this mean that your rule will work?
Repeat step 2 with another face i.e. try replacing with squares or pentagons rather than triangles,
- How does this impact the number of faces, edges and vertices?
Research and explore the following topics:

Further Practice

Some of the essential skills introduced in this lesson are "faces, edges and vertices". You may wish to explore some more regular 3D shapes such as prisms, pyramids, cylinder, cone and sphere. The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.

Transum

Yes No Questions - a game to determine the mathematical item by asking questions that can only be answered yes or no.
Faces, Edges and Vertices - calculate the number of faces, edges and vertices on 3D Shapes.
Faces and Edges - find the number of faces, edges and vertices on some familiar objects.

Extension

Transum

Icosahedron - how many triangles are there on the surface of a regular icosahedron.
The Great Dodecahedron - pupils are not allowed to use their hands to point but must describe fully any shapes they can see in this picture.

Don Steward

Try these 3D geometry: faces, edges and vertices activities by Don Steward.

nRich

Try these Polyhedra activities on nRich.