Chapter 8 Introductory Geometry Section 8.5 Three-Dimensional Geometry

Shapes in three dimensions are sometimes referred to a space figures. The formal characterizations of space figures rely on the concepts of two-dimensional shapes. Lines in Space Two lines that are in the same plane can either intersect or they can be parallel. In three dimensions there is another possibility they can be skew also. Skew lines are lines that are not in the same plane. A B C E D The two lines above picture two skew lines. Neither one sets in a plane with another. A Line and a Plane in Space A line can be either in, out or intersect the plane in just one point. If the line intersects the plane in just one point and is perpendicular to all lines in the plane at the point we say the line and plane are perpendicular.

Planes in Space Just like lines in a plane, planes in space can be either parallel, perpendicular or intersect. The intersection of two different planes is a line. Parallel Planes Perpendicular Planes Intersecting Planes Polyhedra Polyhedrons are three-dimensional shapes whose sides are made up of polygonal regions. There are some examples of polyhedra some non-polyhedra below. polyhedra non-polyhedra

Polygonal Regions A polygonal region is a polygon together with its interior. The vertices of the polygon are also the vertices of the polygonal region. The sides of the polygonal region are referred to as edges. The entire polygonal region is called a face. polygon polygonal region (face) A B C D E 5 vertices: A, B, C, D, E 5 edges: 1 face: ABCDE The polyhedron pictured to the right is called a pentahedron. This is because it has 5 faces. The way a polyhedron is named (i.e. the prefix in front of “hedron”) is determined by the number of faces the solid has. The prefix “penta” means 5 so we have a pentahedron. 6 vertices 9 edges 5 faces

What would you call the shape to the right? It would be called a hexahedron. Prisms vs Pyramids The way that the sides of a polyhedron are arranged also classifies them. A polyhedron that has two identical polygons at opposite faces is called a prism. The two identical polygons are called the bases. A polyhedron that has a polygon as one face so that all of the polygons vertices are connected to the same point is called a pyramid. Here are some examples of different polyhedrons. A pentahedron in the shape of a triangular prism. A hexahedron in the shape of a trapezoidal prism An octahedron in the shape of a hexagonal prism Tetrahedron in the shape of a triangular pyramid Hexahedron in the shape of a pentagonal pyramid

We will reason inductively to get a relationship between the faces (F), vertices (V) and edges (E) of a shape. NAMESTYLESHAPEFVE Tetrahedron Pyramid 446 Pentahedron Pyramid 558 Pentahedron Prism 569 Hexahedron Pyramid 6610 Hexahedron Prism 6812 Heptahedron Pyramid 7712 What is the pattern between F, V and E? F + V = E + 2orF + V – 2 = E

Cylinders and Cones For shapes that do not have polygons for all of there faces can also be classified. cylinders cones