Making a cube part 1

Geometry Level pending

A net for a cube is to be cut from a single unit square of paper. What is the maximum volume, $V$ , of the cube thus formed?

Let $a$ be the least integer such that $V \ge \frac{1}{a}$ . Enter the value of $a$ .

Note: in a proper net, each face is complete and joined to at least one other side along a common edge. (No tricks.)

The answer is 45.

1 solution

Jeremy Galvagni
Sep 17, 2018

Of the 10 nets for a cube, all but one will fit with its sides parallel to those of the square. The side lengths could be $\frac{1}{4}$ , implying $a=64$ . But we can do better with two of the nets if we turn them 45 degrees:

Here the tiltled squares of the net split the unit square into fifths and so the sides of the net squares are $\frac{\sqrt{2}}{5}$ . The volume is then the cube of this.

$(\frac{\sqrt{2}}{5})^{3} \approx 0.0226$ . The reciprocal of this is about $44.194$ so $V \ge \frac{1}{45}$ and $a=\boxed{45}$

Making a cube part 1

The answer is 45.

1 solution

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