Polyhedra: Level 2 Challenges


Given a regular tetrahedron with volume 1cm31 \text{cm}^3 and a cube with volume 1cm31 \text{cm}^3, which object has smaller surface area?

Details and Assumptions:

  • In a regular tetrahedron, all four faces are equilateral triangles, and

  • In a cube, all six faces are squares.

The perfect slice through a cube will reveal a regular hexagonal cross section. If this is a 2 in×2 in×2 in2\text{ in} \times 2\text{ in} \times 2\text{ in} cube, what is the surface area of the hexagonal cross section?

The dodecahedron has 12 pentagonal faces. Therefore it has 12×52=30\frac{12 \times 5}{2} = 30 edges.

The icosahedron has 20 equilateral triangular faces. Therefore it has 20×32=30\frac{20 \times 3}{2} = 30 edges.

How many edges does a truncated icosahedron have?

After drawing in the diagonal perforations onto a unit cube, we are left with a geometric shape object. What is the volume of this new object?

A solid has 12 faces and 20 edges. Given that Euler's Formula applies, how many vertices does it have?


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