Polyhedrons are 3D shapes made of polygons joined at their edges. They have three properties: faces, which are the polygons; edges, where faces meet; and vertices, where edges meet. A convex polyhedron is one whose faces are convex polygons. Euler's Polyhedron Formula states that for any convex polyhedron: F + V - E = 2, where F is the number of faces, V is the number of vertices, and E is the number of edges. This formula is proven using induction and properties of interior and exterior polygon angles.

1. Euler’s Polyhedron Formula and More!

2. Polyhedrons are 3 dimensional figures made
up of polygons, like a cube made up of
squares.
Each polygon making up the polyhedron is
called a face of the polyhedron.
The sides of the polygons where they are
joined together are called the edges of the
polyhedron.

3. The points at which edges meet are called
the vertices of the polyhedron.
Each edge of a polyhedron touches exactly
two faces.
Finally, to be a polyhedron, there must not be
any gaps between the edges or vertices.
Ex. A cube.

4. A convex polygon is a polygon that can be
cut in at most 2 points by a straight line.
If a polygon is not convex, it is concave.
Ex 1. A pentagon is convex.
Ex 2. This is a concave polygon.

5. A convex polyhedron is one whose faces are
convex polygons.
Ex. An tetrahedron is convex, because each
face is a triangle, which is convex.

6. F+V–E=2
F = Number of faces of a polyhedron.
V = Number of vertices of a polyhedron.
E = Number of edges of a polyhedron.
2 = Number after 1.
If you have a convex polyhedron, then the
formula above holds.

7. A cube has 8 vertices, 6 faces, and 12
edges.
8 + 6 – 12 = 2

8. A dodecahedron is a polyhedron composed
of 12 pentagonal faces.
A dodecahedron has 20 vertices, 12
faces, and 30 edges.
20 + 12 – 30 = 2

9. Lemma 1: The interior angles of a convex n-
gon (polygon with n edges) is (n-2)180°.
Proof:
Base Case (n = 3). This is a triangle. The
sum of the angles in a triangle, as we all
know, add up to 180° = (3-2)180°.
Notice that, if you add a side to a polygon,
you can draw a line to make a triangle and
retain your original polygon. Therefore,
adding a side adds 180°.

10. Induction Step (n → n+1). Assume that this
formula holds for an n-gon for some
positive integer n. This means that the sum
of the angles in this n-gon is (n-2)180°. We
want to prove that this implies that this
works for an (n+1)-gon.
Add another side to the n-gon. This adds
180° to the interior angles. So, (n-2)180° +
180°=(n+1-2) 180°, as required.
QED

11. Corollary: The exterior angles of any convex
n-gon adds to 360°.
Proof:
Take any polygon and extend the lines of
each edge out. The line is 180°, but the
inside angle takes away some from that.
Take an n-gon. The sum of the angles of the
lines extended from each edge will be
n180°. (Can you finish this?)
QED