**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**