Doubling the Cube

Classic Problem
Doubling the cube is the exercise to construct a cube whose volume is double that of a given cube.

Solution
Let $C$ be the cube in question.

Let $x$ be the length of one of the edges of $C$.

demonstrated that the problem is equivalent to finding line segments of length $p$ and $q$ such that:
 * $\dfrac x p = \dfrac p q = \dfrac q {2 x}$

from which:
 * $2 x^3 = p^3$

and so:
 * $\dfrac p x = \sqrt [3] 2$

Hence to find a cube whose volume is double that of $C$ is equivalent to finding the Cube Root of 2.

But Doubling the Cube by Compass and Straightedge Construction is Impossible.

Also see

 * Doubling the Cube by Compass and Straightedge Construction is Impossible


 * Trisecting the Angle
 * Squaring the Circle