12.1 exploring solids. polyhedron: three dimensional closed figure formed by joining three or more...

12.1

Exploring Solids

Polyhedron:

• Three dimensional closed figure formed by joining three or more polygons

• Example:

Which of the following are polyhedrons?

No

Yes Yes

No Yes Yes

• Face: sides of a polyhedron that enclose a single region of space

• Edge: a line segment formed by the intersection of two faces

• Vertex: a point where three or more edges meet

Example:

face

edge

vertex

Faces: 6 Vertices: 8 Edges: 12

Identify the number of faces, vertices, and edges for each figure.

Faces- 7Vertices- 10Edges- 15

Faces- 6Vertices- 8Edges- 12

Faces- 5Vertices- 6Edges- 9

Faces- 6Vertices- 8Edges- 12

Faces- 6Vertices- 6Edges- 10

Faces- 8Vertices- 12Edges- 18

Euler’s Theorem:

• The number of faces F, vertices V, and edges E of a polyhedron are related by

F + V – 2 = E

Use Euler’s Theorem to find the unknown number

Faces:

Vertices: 16

Edges: 22

Faces: 5

Vertices:

Edges: 9

Faces:

Vertices: 10

Edges: 15

Faces: 20

Vertices: 12

Edges:

8

6

7

30

• A polyhedron is regular if all of its faces are congruent regular polygons.

• A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.

regular

convex

irregular

concave

• Cross Section: the intersection of the plane and the solid

Describe the shape formed by the intersection of the plane and the

cube

square pentagon triangle