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The major difference between the square and the cube is that a square is a two-dimensional form with just two dimensions (length and width), but the cube is a three-dimensional shape with three dimensions (length, breadth, and height along the x, y, and z-axis).
A cube is a three-dimensional square. A cube has equal values for its x-axis, y-axis, and z-axis when drawn on a cartesian plane. This indicates that all of the sides or measurements of a square are the same. Some real-life examples of cubes are the dice, the shape of the ice we use in our drinks, boxes used for packaging, etc. A cube is a kind of prism that is also known as a square prism.
A cube's volume is the whole three-dimensional space that it occupies. A cube is a three-dimensional solid object with six square faces and all sides of the same length. The cube is one of the five platonic solid forms and is also known as a regular hexahedron. The volume of a cube is expressed as (unit)^{3} or cubic units. The cubic meter (m^{3}) is the SI unit of volume, and it is the volume occupied by a cube with each side length 1m. The volume units used by the United States Centimeter System are inches^{3}, yards^{3}, etc.
The volume of any cube can be determined using two ways: the volume of the cube using sides and the volume of the cube using the diagonals. We will discuss both ways thoroughly.
Let us consider any square-shaped sheet of paper or metal (the choice is yours) with sides equal to ‘k’. Since we are dealing with a square figure, the area swept by the sheet will be its surface area which is the product of the length and width of the square. Now we are aware that the area of a square is a product of its sides; therefore, the area of the square with sides ‘k’ is k^{2}. This was all the 2-dimensional part.
Now let us make a 3D figure. We can construct a 3D figure by providing height to 2D objects; therefore, if we want to design a cube by stacking these sheets of paper such that the total height amounts to a measurement of ‘k’. As a result, the overall space covered by the cube equals the area of the base multiplied by the height. This is known as the volume of a cube and can be formulated as
Volume (V) = k^{3}
We are well aware that the diagonal of any cube with a given side ‘k’ equals √3k. Let us consider the diagonal to be ‘d’; therefore, d = √3k. From this, we get that the side of a cube in terms of diagonal is d/√3.
Putting this value of side into the volume formula, we get:
Note: A typical blunder is to confuse the diagonal of a cube with the diagonal of its face. The face diagonal, on the other hand, is the diagonal on each face of the cube.