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Test Your Geometric Intuition

Visualize the five Platonic Solids--Tetrahedron, Cube; Octahedron, Dodecahedron, and Icosahedron--each of them inscribed in a unit sphere (i.e. sphere with radius of 1 unit).

Visualize their volumes and surface areas, all less than the volume and surface area of the sphere respectively.

Now, based on your visual images, rank them from smallest to largest in terms of their volumes and surface areas.

Question 1: Which Platonic Solid would have the greatest volume? smallest volume?

Question 2: Which Platonic Solid would have the greatest surface area? Smallest surface area?


Source: Adapted from Problem E 2053, American Mathematical Monthly, February 1969.

Hint: Make a guess!

Track down the necessary formulas....Some fancy trig may be involved.

Draw some pictures...Perhaps build some nets for the solids.


Solution Commentary: Surprise...and the winner is the Dodecahedron in both cases:
Solid Side Length Volume Surface Area
Tetrahedron 1.6329... 0.5132... 4.6188...
Cube 1.1547... 1.5396... 8.0000
Octahedron 1.4142... 1.3333... 6.9282...
Dodecahedron 0.7136... 2.7851... 10.5146...
Icosahedron 1.0514... 2.5361... 9.5745...

The calculations have been left to you...or you might enlist the power of some piece of geometrical software.