Critical Geometrical Thinking

The new shape we formed is two identical square pyramids.

The volume of a square pyramid is $\frac{{l^2}h}{3}$ .

Let x represent the side length of the cube.

The side length of our pyramids is $\frac{x{\sqrt{2}}}{2}$ , because the distance from a corner of a square (which is what each face of a cube is) to the center of an adjacent side to the corner is half the side length of the square, and since we're joining the center of one face to the center of a perpendicular face to form a side of our pyramid, we must multiply by $\sqrt{2}$ because of the Pythagorean Theorem.

The height of one of our pyramids is $\frac{x}{2}$ because it is also half the side length of our cube.

The volume of one of our square pyramids is now $\frac{{{(\frac{x{\sqrt{2}}}{2})^2}}(\frac{x}{2})}{3}$ , which simplifies to $\frac{{x^3}}{12}$ .

Since we have two square pyramids, we need to multiply by 2 to get the total volume of the new shape, which gives us $\frac{{x^3}}{6}$ .

The volume of the cube is just $x^3$ . We can now plug both volumes into the ratio to get ${x^3}:\frac{x^3}{6}$ , which simplifies to $\boxed{6:1}$ .