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

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?

• I believe it is a square constructed over that diagonal, i.e. a square which has the cube diagonal as its side. Commented Feb 4, 2019 at 12:01
• Similar terminology is used when Pythagoras' theorem is stated as "the square on the hypotenuse is equal to the sum of the squares on the other two sides". Commented Feb 4, 2019 at 12:10

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

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.