The square on the diagonal of a cube has area 1875. Find a side of the cube and its total surface area.

Question

The square on the diagonal of a cube has an area of 1875 cm$^\text{2}$. Find

1. One side of cube

2. The total surface area of the cube

Moreover, what does ‘square on the diagonal of a cube’ mean?

Answer 1

Let $s$ be the length of the side of either square, q.e. the length of the edge of the cube. Then the area of the square is $s^2$ and therefore the total surface of the cube is $6\ s^2$. And the volume surely would be $s^3$.

By Pythagoras you can calculate the face diagonal via $s^2 + s^2 = 2\ s^2$ to be $\sqrt 2\ s$. Similarily you can calculate the body diagonal of the cube via $(\sqrt 2\ s)^2 + s^2 = 2\ s^2 + s^2 = 3\ s^2$ to be $\sqrt 3\ s$.

From the given value you get $1875\ cm^2 = 3\ s^2$, which is $625\ cm^2 = s^2$. That clearly solves to $s = 25\ cm$.

Now it becomes easy. The total area thus is $6\ s^2 = 750\ cm^2$ and the volume would be $s^3 = 15\ 625\ cm^3$.

--- rk

Answer 2

The square on the diagonal has area $A$ is a fancy way of saying $d^2=A$, where $d$ is the length of the diagonal. It's thought of as a square which has the diagonal as one of its sides.

I think it goes back to when everyone learnt geometry by studying Euclid, which wasn't that long ago. (For example, my father's geometry textbook at school was a version of Euclid's Elements.)

But the traditional wording would probably have been The square on the diagonal is 1875, so they've updated it a bit to make it more strictly logical and to fit the specific problem.