# Diagonal of a Cube

The diagonal of a cube is nearly as simple as the Pythagorean theorem.

It is very simple:

(*length*)^{2}
+ (*width*)^{2}
+ (*height*)^{2}
= (*diagonal*)^{2}

In other words, you simply take the square root of the sum of the squares of the length, width, and height of the cube.

√(*length*)2
+ (*width*)2
+ (*height*)2
= *d*

A simple equation for the diagram above would be this:

√*x*2
+ *y*2
+ *z*2
= *d*

Since the length, width, and height of a cube are equal, you can simply multiply one side squared by 3, like this:

√
3(*x*2)
= *d*

So, if a side of the cube above is equal to 1, then the equation for the diagonal is this:

√
3(12)
= *d*

√ 3
= *d*

It is handy to remember that √ 3 is approximately 1.7 or so. Just keep that tucked away in your memory.

And, if a side of the cube above is equal to 2, then the equation for the diagonal is this:

√
3(22)
= *d*

√ 3(4)
= *d*

2√ 3
= *d*

And, if a side of the cube above is equal to 3, then the equation for the diagonal is this:

√
3(32)
= *d*

√ 3(9)
= *d*

3√ 3
= *d*

You can now see a pattern developing. The diagonal of a cube is equal to any side of the cube times √ 3 .

See whether you can solve the flashcards below.

TAKE YOUR TIME!

- What is the diagonal the cube above?
- What is the area of one face of the cube?5
^{2}or 25 - What is the surface area of the cube?6(25) or 150
- What is the volume of the cube?5
^{3}or 125

- What is the diagonal of the cube above?
- What is the area of one face of the cube?11
^{2}or 121 - What is the surface area of the cube?6(121) or 726
- What is the volume of the cube?11
^{3}or 1,331

- What is the diagonal of the cube above?
- What is the area of one face of the cube?27
^{2}or 729 - What is the surface area of the cube?6(729) or 4,374
- What is the volume of the cube?27
^{3}or 19,683

Assume that the sphere touches all six sides of the cube in the diagram above.

- What's the diameter of the sphere?6
- What is the area of one face of the cube?6
^{2}or 36 - What is the surface area of the cube?6(36) or 216
- What is the volume of the cube?6
^{3}or 216

WAIT! WAIT! Not done yet!

Recall the formula for volume of a sphere:

V = ^{4}/_{3}
π r^{3}

And the formula for surface area of a sphere:

A = 4π r^{2}

- What's the volume of the sphere?
- What's the surface area of the sphere?

Now, try one more diagram. Remember that standardized tests often give answer choices with fractions, roots and π symbols in the answer.

Assume that the sphere touches all six sides of the cube in the diagram above.

- What's the diameter of the sphere?8
- What is the area of one face of the cube?8
^{2}or 64 - What is the surface area of the cube?6(64) or 384
- What is the volume of the cube?8
^{3}or 512 - What's the volume of the sphere?
- What's the surface area of the sphere?