12.1 Exploring Solids
Edge
Basic Definitions
Face
Polyhedra
Polyhedra:
A solid that is bounded by polygons
(called faces) that enclose a single
region of space
Face:
Each polygon that makes up a polyhedra
Edge:
A line segment formed by the
intersection of 2 faces
Vertex:
A point where 3 or more edges meet
Vertex
Tell whether the following is a polyhedra. If it is,
tell how many faces, edges, and vertices it has.
a)
This is a polyhedra
5 faces, 9 edges, 6 vertices
b)
This is a not a polyhedra
Regular Polyhedra: A polyhedra where all faces
are congruent regular polygons
Convex Polyhedra: A polyhedra where any two points on its
surface can be connected by a segment
that lies entirely inside or on the
polyhedran
Decide whether the polygon is regular and/or convex
a)
b)
Convex, regular
Nonconvex, non regular
(or concave)
Cross Section: the intersection of a plane and the solid
Describe cross section
Circle
Eulers Theorem: The number of faces (F), vertices
(V), and edges (E) of a polyhedron are related by the
formula:
F+V=E+2
To find edges: take ½ of the number of sides of
each face
Why? each side is shared by two polygons
a) The solid has 14 faces; 8 triangles and 6 octagons.
How many vertices does the solid have?
F+V=E+2
14 + V = ½[8(3) + 6(8)] + 2
14 + V = 38
V = 24
b) The solid has 5 faces; 4 triangles and 1 square.
How many vertices does the solid have?
F+V=E+2
5 + V = ½ [4(3) + 4] + 2
5 + V = 10
V=5