Chapter 11 Extending Geometry Section 11.4 Three-Dimensional Figures

The Regular Polyhedra A polyhedron is a collection of polygons joined to enclose a region of space. Parts of a polyhedron: Faces Vertices Edges

Platonic Solids Regular polyhedra (Platonic solids) have edges of equal length and the arrangement of polygons at each vertex is the same. Called Platonic solids because Plato associated earth, air, fire, water, and creative energy with these solids and used them in his description of the universe. The ancient Greeks used prefixes that indicated the number of faces to name these polyhedra. Tetra- “four” Hexa- “six” Octa- “eight” Dodeca- “twelve” Icosa- “twenty”

Five Platonic Solids Regular Tetrahedron Regular Hexahedron (Cube) Regular Octahedron Regular Dodecahedron Regular Icosahedron

Five Platonic Solids

Investigation Name of Polyhedron # of Base Edges # of Vertices, V # of Faces, F # of Edges, E F + V

Euler’s Formula Euler’s Formula: F + V = E + 2 Named after Swiss mathematician, Leonhard Euler.

Prisms A prism is a polyhedron with a pair of congruent faces called bases, that lie in parallel planes. The vertices of the bases are joined to form the lateral faces (which are always parallelograms) of a prism. Prisms are named according to the shapes of their bases.

Parts of a Prism

Prisms If the lateral edges of a prism are perpendicular to its bases, the prism is a right prism. If the lateral edges of a prism are not perpendicular to the bases, the prism is an oblique prism.

Right and Oblique Prisms

Pyramids A pyramid is a polyhedron formed by connecting the vertices of a polygon, called the base, to a point not in the plane of the polygon, called an apex. The lateral faces of a pyramid are always triangles. The segment from the vertex perpendicular to the base is called the altitude. Pyramids are named according to the shapes of their bases.

Right Regular Pyramids Right regular pyramid: base is a regular polygon, altitude is perpendicular to base at its center, and lateral faces are isosceles triangles. The height of the isosceles triangular lateral face is called the slant height.

Parts of a Pyramid

Examples of Pyramids

Cylinders, Cones, and Spheres Cylinders, cones, and spheres are all solids with curved surfaces. In a cylinder and cone, the bases are circles. The line through the centers of the bases of a cylinder is called the axis of the cylinder.

Cylinders, Cones, and Spheres

Visualizing Polyhedra A pattern or planar net for a polyhedron is an arrangement of polygons that can be folded to form the polyhedron.

What Polyhedra Can Be Made With These Patterns?