Goal 1: Using Properties of Polyhedra Goal 2: Using Euler’s Theorem CAS 8, 9 12.1 Exploring Solids Goal 1: Using Properties of Polyhedra Goal 2: Using Euler’s Theorem

Polyhedron Edge A solid that is bounded by polygons, called faces, that enclose a single region of space. The plural for polyhedron is Polyhedra face Vertex (Vertices)

Polyhedra vs. Solids Solids are any 3-D figures that enclose a single space. Polyhedra are solids whose faces are all polygons. Ex. A pyramid is a polyhedron, whereas a cone is a solid. See page 719.

Examples of Solids Figure D Figure A Figure C Figure B Figure E

Solids that are also Polyhedra Figure D Figure A Figure C Figure B Figure E

Regular Polyhedron All the faces are congruent regular polygons.

Convex vs. Concave (Nonconvex) A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polygon.

Cross Section The intersection of the plane and the solid. Ex. When you cut an orange in half, the surface of that cut is a “cross section”. It is in the shape of a circle

Platonic Solids There are only five polyhedra whose faces are congruent and all edges are congruent. Tetrahedron- 4 faces Cube- 6 faces Octahedron- 8 faces Dodecahedron- 12 faces Icosahedron- 20 faces Platonic Solid

Theorem 12.1 Euler’s Theorem The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: F + V = E + 2 F - faces V - vertices E - edges

Example 1) A solid has 10 faces and 12 vertices. How many Edges does it have?

Ex) 2 Calculate the number of vertices of the solid using that has 14 faces, 8 triangles, 6 squares