BY DANIEL J. SEBALD Geometric Solids

Introduction Geometric Solids are 3-Dimensional (or “3-D”) shapes – which means they have the 3 dimensions of width, depth, and height. Basic examples are spheres, cubes, cylinders, and pyramids. But there are lots of others. Some geometric solids have faces that are flat, curved, or both. Some have faces that are all the same shape. Some have faces that are different shapes. But they all have 3 dimensions. Sphere Cube Cylinder Pyramid

What are Dimensions? To understand what “3 dimensional” means, you have to understand what a dimension is. The dictionary defines “dimension” as “a measurement of length in one direction.” But that doesn’t really help. It’s easier to understand what it is by looking at examples. A point has no dimensions. A line has 1 dimension because you can only measure it in one direction. A figure drawn on a piece of paper can be measured in 2 directions, so it is 2 dimensional. Finally, a solid can be measured in 3 directions, so it is 3 dimensional.

Types of Geometric Solids: Polyhedra and Non-Polyhedra Solids come in 2 types: polyhedra and non-polyhedra. Non-polyhedra describes any geometric solid that has any surface that is not flat, like a sphere, cone, or cylinder. c ylindercone sphere torus

Polyhedra and Non-Polyhedra Polyhedra describes a geometric solid that has all flat faces, but the faces don’t have to be the same size or shape. Polyhedra must have at least 4 faces but there is no limit to how many faces they can have. Some examples of polyhedra are pictured below: Pentagonal prism Truncated tetrahedron Truncated dodecahedron Truncated cube Pentagonal prism Rhombicuboctahedron

Truncated What? Some of the Polyhedra are called Truncated. Truncated means that something is cut off. In a truncated polyhedra, the corners, called “vertices,” are cut off and replaced with a new face. For example, a truncated cube has new triangle shaped faces where the cube’s vertices were. The shape of the original polyhedra will determine the shape of the new face in each vertex. The pictures below show a cube, and then a truncated cube. Cube Truncated Cube

Platonic Solids A Platonic Solid is a special type of Polyhedra, in which each face is exactly the same, and the same number of faces meet at each corner, or vertex. They were named after a famous philosopher and mathematician from ancient Greece named “Plato.”

Platonic Solids Amazingly, there are only 5 geometric solids that qualify as platonic solids. NAMENumber of Faces Tetrahedron4 Hexahedron(cube)6 Octahedron8 Dodecahedron12 Icosahedron20

The Five Platonic Solids Cube Tetrahedron Dodecahedron Icosahedron Octahedron

Vocabulary Words Dimension: a measurement of length in one direction Edge: the line where faces of a geometric solid meet Face: an individual surface of a geometric solid Non-Polyhedra: a geometric solid that has any surface that is not flat Platonic solid: a polyhedra whose faces are all exactly the same, and the same number of faces meet at each vertex Polyhedra: a geometric solid that has all flat faces Truncate: to cut off the vertices of a geometric sold and replace them with a new face. Vertices: corners of geometric solid (singular is vertex)

Platonic Solids Names _____ icosahedron _____ dodecahedron _____ cube _____ tetrahedron _____ octahedron Quiz Part 1 – Match the platonic solids to their names a.b. c. d.e.

Quiz – Part 2 1. How many sides does an icosahedron have? _________ 2. Is a soccer ball shaped like a a. truncated icosahedron b. truncated dodecahedron c. truncated cube 3. What is a vertex? _____________

Platonic Solids Names __e__ icosahedron __d__ dodecahedron _a___ cube _b___ tetrahedron _c___ octahedron Answers a.b. c. d.e.

Answers 1. How many sides does an icosahedron have? ___20____ 2. Is a soccer ball shaped like a a. truncated icosahedron b. truncated dodecahedron c. truncated cube 3. What is a vertex? __a corner________

Bibliography Algebra & Geometry – Anything But square. Green, Dan Interactives – 3D Shapes January 29, Platonic Solids January 28, Understanding Mathematics. Alfeld, Peter. January 22, Platonic Solids January 29, 2013