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