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icosahedron
Exercise: How big is Your Icosahedron?
Ask the students to measure or calculate how large their icosahedron
is.
This may generate a discussion about what we mean by "how large."
The icosahedron has both a "long diameter" (from one vertex to the
opposite vertex) and a "short diameter" (from one face to the
opposite face). Can the students relate these quantities to inscribed
and circumscribed spheres?
The "long diameter" (widest part) of an
icosahedron may be calculated from the edge length of the icosahedron.
It turns out that the diameter is
times larger than the side length, where
is a constant called the "golden mean." Can
the students derive this quantity (very hard, especially if the
students
aren't familiar with trigonometry!) or experimentally determine the
quantity by measuring the diameters and edge lengths of several
icosahedra?
Can the students calculate the surface area of the icosahedra? What
about obtaining an estimate of volume? Estimating the radius of
inscribed and circumscribed spheres may help the students provide an
upper and lower bound on the volume.
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Previous: Assembling the structure
Next: The Geometry Center's 6-foot
icosahedron
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