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Geometric Surfaces

Problems

Regular Polyhedra and Spheres

How to Cite This SparkNote

Problem : Using your knowledge of concentric circles come up with a definition for concentric spheres.

Concentric spheres are spheres that share a common center.

Problem : Which of the regular polyhedra are prisms?

The only regular polyhedron that is a prism is a cube.

Problem : Triangles can be combined to form any polygon. Is there a geometric solid that can be combined to form any of the regular polyhedra?

Yes. A number of regular pyramids can be combined to form any of the regular polyhedra. The regular pyramids would all be congruent. The bases of the regular pyramids would be the faces of the regular polyhedra. All n regular pyramids (for an n -sided regular polyhedron) would share a common vertex: the point from which all vertices of the regular polyhedron are equidistant. Such regular pyramids, united with their interiors, would form a geometric solid that would be congruent to a regular polyhedron and its interior.

Problem : Based on your knowledge of plane geometry, develop definitions for polyhedra inscribed in a sphere and polyhedra circumscribed about a sphere.

A polyhedron inscribed in a sphere touches the sphere with all of its vertices. A polyhedron circumscribed about a sphere has faces that are all tangent (intersect at one point) to the sphere.

Problem : Devise a definition of the center of a regular polyhedron.

The point from which all the vertices of the polyhedron are equidistant

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