9-4 Geometry in Three Dimensions

Simple Closed Surfaces

Regular Polyhedra

Cylinders and Cones

Simple Closed Surfacesexactly one interior, no holes, and hollow

These two figures are not simple, closed surfaces.

Sphere – the set of all points at a given distance from a given point (the center)

Solid – the set of all points on a simple closed surface together with all interior points

Polyhedron – a simple, closed surface made up of polygonal regions (faces). The vertices of the polygonal regions are the vertices of the polyhedron. The sides of the polygonal regions are called the edges of the polyhedron.

Polyhedra:

Not polyhedra:

Prism – a polyhedron in which two congruent faces (the bases) lie in parallel plane and the other faces (lateral faces) are bounded by parallelograms.

Right prism – lateral faces are perpendicular (bounded) to the bases.

Oblique prism – lateral faces are not perpendicular (not bounded) to the bases.

Pyramid – a polyhedron determined by a polygon and a point not in the plane of the

polygon.

is the Region

or Point

(faces other than the base)

Right pyramid – lateral faces are congruent isosceles triangles.

Regular PolyhedraConvex polyhedron – a polyhedron in which a

segment connecting any two points in the interior of the polyhedron is in the interior of the polyhedron.

Regular polyhedron – a convex polyhedron whose faces are congruent regular polygonal regions such that the number of edges that meet at each vertex is the same for all the vertices of the polyhedron.

Platonic Solids – are regular solid polyhrdra

Cube Tetrahedron Octahedron

Dodecahedron Icosahedron

Nets are plane figure that when folded gives a solid figure. Nets can be used to construct the five regular polyhedras

Example #10a Page 623

Name each polyhedron that can be constructed using the following flattened polyhedra:

Answer:

Right regular hexagonal pyramid. Assume the base is a regular hexagon and the triangles are congruent. Points, if folded up, would meet at a vertex.

Example #10b Page 623

Name each polyhedron that can be constructed using the following flattened polyhedra:

Answer:

Right square pyramid. Assume the base is a square and the triangles are congruent. Points would meet at a vertex.

Example #10c Page 623

Name each polyhedron that can be constructed using the following flattened polyhedra:

Answer:

Cube. All six faces are squares, assume they are congruent.

Example #10d Page 623Name each polyhedron that can be constructed using the following flattened polyhedra:

Answer:

Right square prism. Assume the lateral bases are congruent and the ends congruent. Lateral faces of the prism would all be bounded by rectangles.

Example #10e Page 623Name each polyhedron that can be constructed using the following flattened polyhedra:

Answer:

Right regular hexagonal prism. Assume both bases are congruent regular hexagons and the lateral faces are congruent.

Euler’s Formula

What is the relationship between the number of vertices, faces, and edges of a polyhedron?

Answer: V + F – E = 2

CylindersA simple, closed surface that is not a polyhedron; formed as a line segment AB is parallel to a given line l for a planar curve other than a polygon.

The union of the line segments connecting a point P with each point of a simple, closed curve, the simple, closed curve, and the interior of the curve.

Cones

HOMEWORK 9-4

Page 623 - 627

7, 9, 11, 13, 15, 23

CHAPTER REVIEWPage 639 - 641

1, 3, 5, 7, 11, 15, 17, 19