## Friday, December 22, 2006

### Geometry 12.1 notes

12.1 Exploring solids

Polyhedron - is a solid that is bounded by polygons called faces that enclose a single region of space.

An edge of a polyhedron is a line segment formed by the intersection of two faces.

A vertex of a polyhedron is a point where three of more edges meet.

Plural = polyhedra

Example:

Regular Polyhedron - all faces are congruent and all the faces are regular polygons (polygons that have all angles congruent and all sides congruent).

Convex Polyhedron - any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.

Cross-section - intersection of the plane and the solid. pg. 720

Platonic Solids -

Regular Tetrahedron (4 faces, 4 vertices, 6 edges) - associated with the element of fire.

Cube (6 faces, 8 vertices, 12 edges) -

Regular Octahedron (8 faces, 6 vertices, 12 edges) - associated with the element of air.

Regular Dodecahedron (12 faces)

Regular Icosahedron (20 faces, 12 vertices, 30 edges) - associated with the element of water. A soccer ball’s pattern is a truncated icosahedron.

Euler’s Theorem:

The number of faces (F), vertices (V), and edges (E), of a polyhedron are related by the following formula:

F + V = E + 2

Example: a polyhedron has 20 faces; all triangle

Since a triangle has 3 sides and each of the triangles are going to share an edge, you have

20 + V = ((3)(20))/2 + 2

20 + V = (60)/2 + 2

20 + V = 30 + 2

20 + V = 32

V = 12

Homework: #40 pg. 723 #10-30 even, 32-35, 54, 55, 60-62, 69