Download - Section 9.4 Polyhedrons & Spheres
Section 9.4 Nack/Jones 1
Section 9.4Polyhedrons & Spheres
Section 9.4 Nack/Jones 2
Polyhedron
• Plural: polyhedrons or polyhedra• A solid bounded by plane regions.• The faces of the polyhedrons are polygons• The edges are the line segments common to these
polygons • Vertices are the endpoints of the edges• Convex: Each face determines a plane for which all
remaining faces lie on the same side of the plane. p.434.• Concave: Two vertices and the line segment containing
them lies in the exterior of the polyhedron.
Section 9.4 Nack/Jones 3
Euler’s Equation
• Theorem 9.4.1: The number of vertices V, the number of edges, E, and the number of faces F of a polyhedron are related by the equation.
V + F = E + 2
Where V = # of vertices
F = # of faces
E = # of edges
Example 1 p. 434
Section 9.4 Nack/Jones 4
Regular Polyhedron
• A regular polyhedron is a convex polyhedron whose faces are congruent regular polygons arranged in such a way that adjacent faces form congruent dihedral angles (the angle formed when two edges intersect).
Ex 2 p. 435
Section 9.4 Nack/Jones 5
Spheres
• Three Characteristics
1. A sphere is the set of all points
at a fixed distance r from a given
point O. Point O is known as the
center of the sphere.
2. A sphere is the surface determined when a circle
(or semicircle) is rotated about any of its diameters.
3. A sphere is the surface that represents the theoretical
limit of an “inscribed” regular polyhedron whose number
of faces increase without limit.
Section 9.4 Nack/Jones 6
Surface Area and Volume of a Sphere
• Theorem 9.4.2: The surface area S
of a sphere whose radius has length r
is given by S = 4r²
Ex. 3 p. 437• Theorem 9.4.3: The volume V of a
sphere with radius of length r is given by V =4/3 r3
Example 4 – 6 p. 429• Solids of Revolution: Figures p. 439
– Revolving a semi circle = sphere– Revolving circle around line = torus