Determine the Volume of Space the Astronaut Can Visit Around the Cube

An astronaut is tethered to a vertex of a cube with a side length of $3$ meters. What is the total volume of the space (in cubic meters) that he is able to visit if the length of the rope is $4$ meters?

Bonus: What if the rope length is $3(1 + \sqrt{2})$ meters?

The first part of the problem was originally asked here.

The second part of the problem is very interesting since the rope now extends to the opposite vertex. It is clear that the minimum distance between two opposing vertices is $3\sqrt{5}$ , so the remaining length is $3(1 + \sqrt{2} - \sqrt{5})$ . But does that necessarily mean the astronaut can go that far around the opposing vertex?

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$