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Browse College Polyhedra

Stars indicate particularly interesting answers or good places to begin browsing.



Definitions as a Tool of Mathematics [03/04/2003]
Defining polyhedra: regarding unusual polyhedra, how can we exclude 'doubled' shapes, e.g. two tetrahedra that join at a single vertex (sort of like an hourglass)?

Biggest Cuboid in a Sphere [03/20/2003]
I have a sphere with radius of 10cm. I have to find the biggest cuboid that fits in that sphere.

Calculating Angles Between Faces of a Solid [09/15/2003]
How can I compute the dihedral angles for a Great Rhombicosidodecahedron?

Cylinders and Euler's Rule [09/06/2002]
How and why does Euler's rule work for cylinders?

Euler's Formula for Polyhedra [08/12/1997]
How would you prove Euler's formula V-E+F = 2 for all polyhedra of genus zero?

Faces, Vertices, and Edges of Cylinders, Cones, and Spheres [12/28/2003]
Do cylinders, cones and spheres have faces, vertices, and edges? I'm not sure how they would fit into Euler's formula of v - e + f = 2.

Finding the Area of an Irregular Polygon [02/23/2008]
What is the formula for finding the area of an irregular polygon?

Finding the Area of an Irregular Polygon in 3-D [02/04/2009]
Given the coordinates of eight points in three-dimensional space, with no known angles, how can I find the area of the polygon they define?

Find the Edge Lengths of a Cuboid [05/13/2003]
I am trying to discover the lengths of the edges of a cuboid when only the diagonal, area, and volume are known.

Geodesics [12/15/1996]
Can you give me information on the math behind geodesics?

Height of Tetrahedral Pyramid [09/12/2001]
I'm looking for a simple formula (and derivation) of the height of a tetrahedral pyramid with an equilateral triangle as a base.

Higher-Dimensional Cubes [07/04/2003]
I've written a program that draws cubes in any number of dimensions between 0 and 15. I want to have it display the number of faces and cells of each dimension each cube has.

Maximum Surface Area for Total Edge Length [07/14/2002]
A piece of wire of total length L units is used to form the nine edges of a prism whose ends are equilateral triangles and whose other faces are rectangles. What is the maximum surface area of this prism?

Mitres on Pyramids [09/26/2002]
I am weatherproofing my home, and have to mitre boards in a pyramid with a rectangular - not square - base, and an apex that is directly over the centre of one edge of the base.

Monogons and Digons - Polygons with Fewer Than 3 Sides [01/24/2006]
What do monogon and digon polygons look like? How can you have a polygon with fewer than three sides?

Orbit and Stabilizer in Rotational Symmetry [11/11/2004]
Calculate the orders of the following groups of rotations: of a regular tetrahedron, a regular octahedron, a regular dodecahedron, and a regular icosahedron. I'm having trouble figuring out the stabilizers. I know that the order of the group of rotations is equal to the order of the orbit times the order of the stabilizer.

Orthocentric Tetrahedron [11/30/2002]
Recall that the opposite edges of an orthocentric tetrahedron are perpendicular. Let ABCD be an orthocentric tetrahedron. Show that AB^ 2 + CD^2 = AD^2 + BC^2.

Platonic Solids [1/31/1996]
What are the Platonic solids in 4 and more dimensions?

Polyhedra: Solids or Surfaces? [06/06/2003]
Is a polyhedron always a solid figure?

Polyhedron Problem [10/29/1996]
How many faces share each edge?

Polyhedron Project [10/29/2003]
I'd like to do a project involving tetrahedra. What would you suggest?

Spheres in a Cylinder [05/20/2003]
Spheres of 6 cm radius are dropped into a cylinder of radius 8 cm and height 36 cm. How many spheres can fit into the cylinder?

Spiral Inside a Hexagonal Room [09/03/2003]
Two walls meet at 120 degrees, and you have a piece of cardboard with an angle of 137.5 degrees that you want to tilt until its sides are snug against the wall. How do you find the angle of tilt?

Tetrahedron Projected on a Plane [10/29/1996]
How do you project a regular tetrahedron perpendicularly onto a plane to get the maximum area shadow?

Understanding Fourth Dimension Figures [07/05/1998]
Can you help me figure out the equations for fourth dimension figures such as the tesseract and the hypertetrahedron?

Volume of a Soccer Ball [05/31/2003]
What is the simplest way to find the volume of a truncated icosahedron (aka soccer ball or buckyball)?

Volume of a Tetrahedron [01/23/2002]
The volume of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face. Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S.

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