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.

Various techniques were devised which used more elaborate apparatus than just the straightedge and compass:

Also see

 * Doubling the Cube by Compass and Straightedge Construction is Impossible


 * Trisecting the Angle
 * Squaring the Circle