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Show that (a) the total surface of a cube is twice the square of its diagonal, (b) the volume of a cube is 1/9 √3 times the cube of its diagonal.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
Consider the lower base of a cube. Use Pythagorean Theorem in order to solve for the hypotenuse or the diagonal of the lower base as follows
If all the faces of a cube are perpendicular to each other, then all edges are perpendicular to each other also. Since c is located at the lower base of a cube, then c is perpendicular to x.
Use Pythagorean Theorem in order to solve for the diagonal of a cube as follows
The total area of a cube is
The volume of a cube is
Therefore,
(a)
(b)
