Figure 1

A regular polygon is a polygon whose angles are
equal and side lengths are equal. A 3D polyhedron is
said to be regular if all its faces are regular
polygons and the same number of faces meet at each corner.
How many regular 3D solid polyhedra are there?
Answer:there are exactly five
the tetrahedron (pyramid with triangular faces),
octahedron (8sided with triangular faces),
dodecahedron (12sided with pentagonal faces),
icosahedron (20sided with triangular faces),
and cube (6sided with square faces).
These are sometimes called the Platonic Solids, named thus
because they were described by Plato in one of his books.
Euclid proved that there are only five.
Presentation Suggestions:
See if students can think of (or construct)
some other polyhedra whose faces are the same shape,
but allowing nonregular faces and/or
different numbers of faces to meet at corners.
The Math Behind the Fact:
A regular solid with hexagonal faces cannot exist because
if it did,
the sum of the angles of any 3 hexagonal corners that
meet would already equal 360,
so such an object would be planar.
Thus the only regular solids involve pentagons,
squares, and triangular faces and you can further limit the
possibilities using the
Euler characteristic of the sphere, which is 2.
This makes a great exercise.
How to Cite this Page:
Su, Francis E., et al. "Regular Solids."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
