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12.1
Exploring Solids
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Polyhedron:
• Three dimensional closed figure formed by joining three or more polygons
• Example:
![Page 3: 12.1 Exploring Solids. Polyhedron: Three dimensional closed figure formed by joining three or more polygons Example:](https://reader036.vdocuments.site/reader036/viewer/2022081809/5697c00b1a28abf838cc7fe6/html5/thumbnails/3.jpg)
Which of the following are polyhedrons?
No
Yes Yes
No Yes Yes
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• 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
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Example:
face
edge
vertex
Faces: 6 Vertices: 8 Edges: 12
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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
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Euler’s Theorem:
• The number of faces F, vertices V, and edges E of a polyhedron are related by
F + V – 2 = E
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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
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• 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
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• Cross Section: the intersection of the plane and the solid
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Describe the shape formed by the intersection of the plane and the
cube
square pentagon triangle