Doubling the Cube

Unfortunately, the answer is that it can not be solved with only a compass and a straightedge. It CAN be solved with additional tools some of which can be constructed using a ruler and a straightedge but not without cutting it out of a sheet of paper.

To double the cube means to be given a cube of side length s and volume V, and to construct a new cube, larger than the first, with volume 2V and therefore side length . The problem is known to be impossible to solve with only compass and straightedge, because is not a constructible number.

The straightedge and compass give you the ability to produce ratios which are solutions to quadratic equations, but doubling the cube and trisecting the angle require ratios which are the solution to cubic equations, while squaring the circle requires a transcendental ratio.

A ruler (which is not a straightedge due to the inclusion of a marked unit of distance) solves the problem - but breaks the rules of classical construction as a ruler is clearly a third tool. 

Figure 1:  You can only do it if you cheat.

With a piece of paper, construct a ruler with a single unit distance marked on it. Construct an equilateral triangle ABC with side length 1, and extend side AB by one unit to form the line segment ABD. Extend side BC to form the ray BCE, and draw the ray DCF. Now take the ruler and place it such that it passes through vertex A and intersects DCF at G and BCE at H, such that the distance GH is exactly 1. The distance AG will then be precisely.