Three-dimensional shapes, or solids, can be constructed by joining together flat or curved
shapes in various ways.
The Platonic solids are made up of all the same shapes.
Prisms and pyramids are made up of a flat base that either extends all the way through
the solid in a prism, or tapers evenly to a point at the apex of a pyramid.
And then there are other shapes made from curved surfaces of varying complexity.
Most of these shapes shown here are polyhedral.
A polyhedron is a three-dimensional solid made up entirely of flat faces joined along
straight edges.
The cylinder, cone, sphere and torus all have at least one curved face, but each of the
other shapes here is a polyhedron.
A Platonic solid, named after the ancient Greek philosopher Plato, is a convex polyhedron
whose faces are all exactly the same shape and size, and with the same number of faces
meeting at every vertex.
Although it may seem like you could make any number of different-shaped solids meeting
these criteria, in fact there are only five of them.
There are only five ways to make a Platonic solid.
The tetrahedron is made from four equilateral triangles.
It’s a triangular-based pyramid.
Hmm, can you calculate the vertical height of a tetrahedron?
It’s not as easy as it sounds.
The cube has six square faces.
It’s a square-based prism whose height, width and length are equal.
The octahedron has eight triangular faces.
It, and the rest of the Platonic solids, are neither prisms nor pyramids.
The dodecahedron has twelve pentagonal faces.
“Do-deca-” means two and ten.
And the icosahedron again has triangular faces: twenty of them.
The tetrahedron has three triangles meeting at each vertex; the octahedron four; and the
icosahedron five.
Can you work out why there are only five Platonic solids?
Look up Euler’s formula and see if you can prove it to yourself.
A prism is a solid shape with a flat base that keeps the same cross-section all the
way through its length.
If you take a cross-section parallel to the base, wherever you look along the prism you’ll
see exactly the same shape and size all the way along.
Be careful though, the “base” may not actually be on the bottom.
In this prism, if I try to use the bottom surface, I’ll get a rectangular cross-section
that gradually reduces in width but not length, until it becomes a line at the top.
But if instead I use the triangular end, look what happens: It’s the same triangle all
the way along.
A pyramid, on the other hand, has the same shape cross-section, but it gradually becomes
smaller and smaller until it reaches a single point.
That point is called the apex of the pyramid.
Again, the base may not always be the one on the bottom.
Let’s try classifying a few shapes as prisms or pyramids.
This tetrahedron is a pyramid, because whichever face you choose, if you take cross-sections
parallel to that face, you’ll get equilateral triangles getting smaller and smaller until
you reach the opposite vertex.
It’s a triangle-based pyramid.
Likewise, this cube is a prism, because again, you can choose any face you like; the cross-section
will stay exactly the same all the way through.
It’s a square-based prism.
This one is a triangle-based prism.
As I showed you just before, you have to be careful to select the correct face as the
base.
Here you need to choose one of the triangular faces on the ends.
Hmm, what about this one?
Well, there’s quite a few faces meeting at one point; do you think that might be the
apex of a pyramid?
Let’s try it.
Choose the opposite face as the base, and take cross-sections parallel to that base.
Yep, the same shape all the way along, tapering to a single point at the apex.
It’s an irregular-pentagon-based pyramid.
Can you work this one out?
Choose carefully which side should be the base.
This side won’t work.
Nor will this one.
Ah, but this one does.
It’s an irregular-pentagon-based prism.
This one should be easier now you’ve got the idea.
Can you see the base?
It’s getting smaller to a point, so it’s a pyramid.
What shape is that cross-section?
I see two parallel sides.
It’s a trapezium.
This is a trapezium-based pyramid.
Now, what about a cylinder?
Some people say it’s not a prism because one of its faces is curved.
Well it’s certainly not a polyhedron, because a polyhedron has to have all flat faces.
But by our definition, a cylinder is a prism: It has a flat base, the circle on the end,
and if we look at a cross-section parallel to that end, we see the same circle all the
way along.
That means it’s a circle-based prism.
Similarly, a cone is not a polyhedron, because it has a curved face.
But it is a pyramid.
Again, take the circle as the base.
There’s a circular cross-section, getting smaller and smaller to single point at the
apex.
It’s a circle-based pyramid.
Now a tricky one.
I said earlier that this shape, an octahedron, is neither a prism nor a pyramid.
Look at what happens if I take a cross-section parallel to one of the faces, and look at
slices all the way through the shape.
The same thing will happen whichever face you choose.
This is neither a prism nor a pyramid.
And there are plenty of other shapes like this.