Geometric Solids
BY DANIEL J. SEBALD
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.
cylinder
sphere
torus
cone
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 cube
Truncated dodecahedron
Truncated tetrahedron
Rhombicuboctahedron
Pentagonal prism
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
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
Platonic Solids
Amazingly, there are only 5 geometric solids that
qualify as platonic solids.
NAME
Number of Faces
Tetrahedron
4
Hexahedron(cube)
6
Octahedron
8
Dodecahedron
12
Icosahedron
20
The Five Platonic Solids
Cube
Dodecahedron
Tetrahedron
Octahedron
Icosahedron
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)
Quiz
Part 1 – Match the platonic solids to their names
Names
Platonic Solids
a.
b.
c.
d.
e.
_____ icosahedron
_____ dodecahedron
_____ cube
_____ tetrahedron
_____ octahedron
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?
_____________
Answers
Names
Platonic Solids
a.
b.
c.
d.
e.
__e__ icosahedron
__d__ dodecahedron
_a___ cube
_b___ tetrahedron
_c___ octahedron
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. 2011.
Interactives – 3D Shapes
www.learner.org/interactives/geometry
January 29, 2013.
Platonic Solids
www.mathisfun.com/platonic_solids.html
January 28, 2013.
Understanding Mathematics.
Alfeld, Peter. January 22, 1997.
Platonic Solids
http://mathworld.wolfram.com/platonicsolid.html
January 29, 2013