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Questions and Answers
What is a characteristic of a regular polyhedron?
What is a characteristic of a regular polyhedron?
What is the main difference between a convex polyhedron and a concave polyhedron?
What is the main difference between a convex polyhedron and a concave polyhedron?
Which of the following is not a polyhedron?
Which of the following is not a polyhedron?
How many regular polyhedra are there?
How many regular polyhedra are there?
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What is the term for a polyhedron that is not regular?
What is the term for a polyhedron that is not regular?
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What is the defining characteristic of a concave polyhedron?
What is the defining characteristic of a concave polyhedron?
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What is Euler's characteristic, and what does it describe?
What is Euler's characteristic, and what does it describe?
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What is a 'net' in the context of polyhedron models?
What is a 'net' in the context of polyhedron models?
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What is a use of polyhedron models in computer-generated imagery (CGI)?
What is a use of polyhedron models in computer-generated imagery (CGI)?
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What type of polyhedron has 20 faces?
What type of polyhedron has 20 faces?
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Study Notes
Polyhedra
- A polyhedron is a three-dimensional solid with flat faces, examples include cubes, pyramids, and prisms.
- Spheres and cylinders are not polyhedra because they have curved faces.
Types of Polyhedra
- A regular polyhedron (also known as a Platonic solid) is a polyhedron with all faces being regular polygons that are congruent to one another.
- There are exactly five regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
- These five polyhedra are very symmetrical and are also known as Platonic solids.
Convex Polyhedra
- A convex polyhedron is a polyhedron in which all vertices "go out" and there are no "dimples" or "dents".
- Formally, a convex polyhedron is one where a line segment connecting any two points on the polyhedron is entirely within the polyhedron.
Concave Polyhedra
- A concave polyhedron is a polyhedron with one or more vertices, edges, or faces that "go into" the polyhedron.
- Formally, a concave polyhedron is one where a line segment connecting two points on the polyhedron also contains points outside the polyhedron.
Polyhedron Formula
- Euler's characteristic states that the numbers of faces (F), vertices (V), and edges (E) of any convex polyhedron satisfy the equation: F + V = E + 2.
Polyhedron Models
- A polyhedron model is a representation of a polyhedron, which can be physical or created using computer graphics.
- Physical models can be constructed by cutting out a "net" of the polyhedron and then folding and gluing it together.
- Computer-generated models can be rotated, expanded, and colored to make them easier to understand.
Uses of Polyhedron Models
- Polyhedron models are used in computer-generated imagery (CGI) and in architecture, where buildings are typically polyhedral.
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Description
Learn about polyhedra, a three-dimensional solid with flat faces, including types such as cubes, pyramids, and prisms. Discover the differences between polyhedra and curved shapes like spheres and cylinders.
