# ISEE Upper Level Quantitative : How to find the diagonal of a cube

## Example Questions

### Example Question #1 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with a side length of  ? Round to the nearest hundreth.

Explanation:

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions).  We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula.  This is very easy since one point is all s.  It is merely:

This is approximately .

### Example Question #2 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with a side length of  ? Round to the nearest hundreth.

Explanation:

It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula.  This is very easy since one point is all s.  It is merely:

This is approximately .

### Example Question #3 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with a volume of  ? Round to the nearest hundredth.

Explanation:

First, you need to find the side length of this cube. We know that the volume is:

, where  is the side length.

Therefore, based on our data, we can say:

Solving for  by taking the cube-root of both sides, we get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .

### Example Question #4 : How To Find The Diagonal Of A Cube

What is the length of the diagonal of a cube with a surface area of  ? Round your answer to the nearest hundredth.

Explanation:

First, you need to find the side length of this cube. We know that the surface area is defined by:

, where  is the side length. (This is because the cube is  sides of equal area).

Therefore, based on our data, we can say:

Take the square root of both sides and get:

Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:

(Note that not all points are drawn in on the cube.)

The two points we are looking at are:

and

Solve this by using the distance formula. This is very easy since one point is all s. It is merely:

This is approximately .