Solid Geometry

1. Solid Geometry • Space is the set of all points • The geometry of objects in space is called solid geometry • Objects in space are three dimensional • When drawing three dimensional figures, hidden lines are drawn using dashes • A polyhedron is a solid made up of several polygonal surfaces • Each surface is a face of the polyhedron • An edge is the line segment where two faces meet • Just as polygons must only have straight sides, polyhedra must only have polygonal faces • Polyhedra are named based on the number of faces • For example, an heptahedron has seven faces • A tetrahedron is a polyhedron with four faces • A cube is a hexahedron with square faces

2. Solid Geometry • There are five Platonic solids in which the faces are congruent regular polygons • Any cube, which by definition has 6 square faces • A tetrahedron with 4 equilateral triangles as faces • An octahedron with 8 equilateral triangles as faces • An icosahedron with 20 equilateral triangles as faces • A dodecahedron with 12 regular pentagons as faces • We will focus on two specific types of polyhedra • A prism, which has two congruent parallel bases and rectangular faces • The prism at right is a pentagonal prism • A pyramid, which has a single polygonal base and triangular faces that meet at a common vertex • The pyramid at right is a hexagonal pyramid

3. Solid Geometry • A sphere is the set of all points in space equidistant from a center point • A hemisphere is a sphere cut in half by a plane passing through its center • Other common non-polyhedral solids • A cylinder is like a prism with circular bases • A cone is like a pyramid with a circular base • A solid of revolution is created by rotating a two dimensional shape along an axis • Rotating a rectangle along a side creates a cylinder • What shape would be created by rotating a right triangle along one of its legs?

4. Solid Geometry • A cross-sectionis a two dimensional shape created by slicing a solid with a plane • The cross-section of a sphere is a circle • A cross-section of a cube could be a square, a rectangle, a triangle, or even a hexagon • A cross-section of a cone could be a circle, an ellipse, a triangle, a parabola, or a hyperbola • The properties of these conic sections are quite useful, and are studied in later math courses