There are 7.48 gallons in cubic foot. Set up a ratio:

1 ft³ / 7.48 gallons = x cubic feet / 10,000 gallons

Pool Volume = 10,000 gallons = 10,000 gallons * (1 ft³/ 7.48 gallons) = 1336.9 ft³

Pool Volume = 4ft x 10 ft x WIDTH = 1336.9 cubic feet

Solve for WIDTH:

4 ft x 10 ft x WIDTH = 1336.9 cubic feet

WIDTH = 1336.9 / (4 x 10) = 33.4 ft

The cube has a volume of 64cm³, making the length of one edge 4cm (4 * 4 * 4 = 64).

So the area of one side is 4 * 4 = 16cm²

The formula to find the volume of the cube is

Since we know the volume, we can set up the equation

To find the length of the side of a square given the surface area, use the surface area formula and solve for :

, now divide both sides by 6
, now square root both sides
.

To find the length of the edge of a cube from its surface area, remember that , where  is the length of a side.

So, the box is  units long.

To find side length, simply realize that volume is the side length cubed. Thus,

To solve, simply take the cube root of the volume. Thus,

We begin with a picture, noting that the diagonal, labeled as , is the length across the cube from one vertex to the opposite side's vertex.

However, the trick to solving the problem is to also draw in the diagonal of the bottom face of the cube, which we labeled .

Note that this creates two right triangles.  Though our end goal is to find , we can begin by looking at the right triangle in the bottom face to find .  Using either the Pythagorean Theorem or the fact that we have a 45-45-90 right traingle, we can calculate the hypotenuse.

Now that we know the value of , we can turn to our second right triangle to find  using the Pythagorean Theorem.

Taking the square root of both sides and simplifying gives the answer.

Recall that the volume of a cube is computed using the equation

, where  is the length of one side of the cube.

So, for our data, we know:

Using your calculator, take the cube root of both sides. You can always do this by raising  to the  power if your calculator does not have a varied-root button.

If you get , the value really should be rounded up to . This is because of calculator estimations. So, if the sides are  , you can find the diagonal by using a variation on the Pythagorean Theorem working for three dimensions:

This is . Round it to .

Recall that the diagonal of a cube is most easily found when you know that cube's dimensions. For the volume of a cube, the pertinent equation is:

, where  represents the length of one side of the cube. For our data, this gives us:

Now, you could factor this by hand or use your calculator. You will see that  is .

Now, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem / distance formula:

 or 

You can rewrite this: