The Platonic Solids
Discovery 1: How many Platonic Solids exist in this world?
Materials:
Regular geometric tiles (i.e., equilateral triangles, squares, pentagons, and hexagons)
The Challenge: Construct a 3D corner (vertex) using only one type of regular polygon. How many can you find?
Concept focus:
A Platonic Solid is a 3D shape where
- each face is the same regular polygon
- the same number of polygons meet at each vertex (corner)
- The total angle at each vertex must be less than 360°
Discovery 2: Pattern Recognition
Group Activity:
- Each group builds a Platonic solid using recycled materials (i.e., straw for edges, cardboard for faces)
- Create a table of values that show the number of vertices, edges, and faces for each solid.
Tetrahedron
Hexahedron / Cube
Octahedron
Dodecahedron
Icosahedron
Properties & Conditions for Building Platonic Solids
Polyhedron Criteria:
- Euler’s Characteristic - The solid must satisfy the formula V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces.
- Face regularity - All faces must be congruent regular polygons.
- Vertex regularity - The same number of faces must meet at every vertex in exactly the same arrangement Convexity - The solid must be convex (line segment test); If you pick any two points inside the shape, the line connecting them stay entirely inside the shape
- The 360o Limit - If you sum the interior angles of the polygons meeting at a vertex, the total must be less than 360o .
Solution
Discovery 1: Only 3, 4, or 5 triangles, 3 squares and 3 pentagons work. If you join 6 triangles or 3 hexagons, it creates a flat 360o surface, so they cannot fold into a solid. Which lead to a conclusion why only five exist.
Discovery 2: It is impossible to have more than 5 platonic solids, because any other possibility violates or does not satisfy the number of edges, corners and faces according to Euler's Formula.
1. Verify if the following solid is a Platonic solid.
1. Verify if the following solid is a Platonic solid.
3. Verify if the following solid is a Platonic solid.
4. Verify if the following solid is a Platonic solid.
Duality in Platonic Solids
Materials: 3D models / Diagrams of the 5 Platonic Solids
Perform the following steps for each solid:
- Mark the center point of every face.
- Imagine or physically connect using string/wire/ drawing these center points to form a new internal solid.
- Compare the new solid formed inside the initial solid.
- vertices & faces: the number of faces of the initial solid equals the number of vertices of the new, and vice-versa (i.e., a Cube with 6 faces & 8 vertices creates an Octahedron with 8 faces & 6 vertices inside it).
- edges: dual pairs have the same number of edges.
- a solid can be perfectly inscribed inside its dual partner, where the vertices touch the centers of the faces of the outer solid.
Surface Area
Use the fundamental geometry of a regular -gon and build the formula.
For any regular polygon with sides of length , the area of the polygon is given by
Area of polygon
For a triangle (), this simplifies to area = .
Find the surface area for a tetrahedron, cube, octahedron, dodecahedron, and icosahedron with length .
Volume
The volume formula is derived from the idea that any Platonic solid can be divided equally into equal pyramids.
Each pyramid has
- base = 1 face of the solid
- apex = center of the solid
- height = distance from the center of the solid to the center of the face