Year 10 Advanced Mathematics or Convex Regular Solids PLATONIC SOLIDS More correctly known as 

Solids can be broken up into 2 groups  What are Platonic solids?  Prisms Let’s quickly revise solids in general and Pyramids Then the names depend on the...number of faces none Pentagonal Prism Hexagonal based Pyramid Hexagonal Prism Heptagonal based Pyramid Octagonal based Pyramid Heptagonal Prism faces Nonagonal based Pyramid Octagonal Prism Triangular based pyramid none Square or Rectangular based Pyramid none cube Pentagonal based Pyramid Rectangular Prism & And so on for the PyramidsAnd so on for the Prisms But which ones of these are Platonic???  Prisms Pyramids

Q. What does it mean to be a Platonic solid?  A. Each of the faces must have the same length sides AND... All the faces in the solid must be the same. Q. Which ones satisfy our 2 criteria so far??  In the prisms, the only one that does is the….. And in the pyramids, the only one that does is the….. Cube Triangular based pyramid There must be more!

Q. Why are they the first two Regular solids?  A. The first shape for each face of any Regular solid must have 3 sides and the next shape must have 4 sides on each of its faces. Q. What are the special names that are used for each of these solids? A. The one with 3 sides is called a  …… Triangular Pyramid and the one with 4 sides is called a  …… Cube Both of them have other names……

The Cube is also called a Hexahedron The Triangular Pyramid is also called a Tetrahedron What other polygons could our regular solids have as their faces? Octagons Hexagons Pentagons Both these names come from the Greek language. Hedron means surfaces & the prefixes say how many. Heptagons Nonagons Decagons

All solids are made up of flat surfaces which meet at both their edges and their vertices. Q. How many faces MUST at least meet at each vertex? Look at the solids you have made……… 3 How many regular polygons can meet at each vertex? Which regular polygons will not make a solid? Some questions to help you decide the other regular solids…… What part of each of the polygons tells you whether it will work to make a regular solid or not? Now, ….

The answers to our questions for the other regular solids were…… How many regular polygons can meet at each vertex? Which regular polygons will not make a regular solid? What part of each of the polygons tells you whether it will work to make a regular solid or not? 3 or 4 All polygons with more than 5 sides The angles. The last answer gives us another question…… What must the sum of the angles of the polygons at any vertex be less than? 360° Cut out the diagrams on the second sheet to be able to answer these questions. Put them together with the cut-out from the person next to you.

So, which regular shapes will work to make regular convex vertex points?  Triangles and Squares Pentagons When you used 3 equilateral triangles meeting at a vertex, you had the start of a  Tetrahedron When you used 4 equilateral triangles meeting at a vertex, you had the start of an  Octahedron When you used 5 equilateral triangles meeting at a vertex, you had the start of an  These are the only convex regular polyhedra with equilateral triangles as their faces.  because 3 equilateral triangles total 180° at a vertex because 4 equilateral triangles total 240° at a vertex because 5 equilateral triangles total 300° at a vertex Icosahedron

What about Squares and Pentagons?  It was only possible to use 3 squares meeting at a vertex. You had the start of a  Hexahedron (cube) With the Pentagons, your diagram from the investigation sheet only had 3 of them, giving a total at any vertex of 3  108° = 324°. It was the start of a Dodecahedron It was impossible to use 3 hexagons because each has an angle of 120°, already giving a total of 360° These are the only other convex regular polyhedra with regular shapes as their faces. because 3 squares total 270° at a vertex

So, there are 5 Platonic Solids. They are the: Icosahedron Octahedron Dodecahedron Tetrahedron Hexahedron Cut out the other nets for the other 3 Platonic solids and complete the table on the worksheet