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When shapes get three-dimensional, things get weird. Dive in to learn about the cube, tetrahedron, octahedron, and more.

Given a regular tetrahedron with volume \(1 \text{cm}^3\) and a cube with volume \(1 \text{cm}^3\), 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.

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

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

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

**How many edges does a truncated icosahedron have?**

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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?

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