# Diagonal of a Square

The diagonal of a square is as simple as the Pythagorean theorem.

It is very simple:

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

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

√(*length*)2 + (*width*)2 = *d*

A simple equation for the diagram above would be this:

√*x*2 + *y*2 = *d*

Since the length and width of a square are equal, you can simply multiply one side squared by 2, like this:

√ 2(*x*2) = *d*

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

√ 2(12) = *d*

√ 2 = *d*

Incidentally, √ 2 is approximately 1.4. That’s a handy number to remember.

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

√ 2(22) = *d*

√ 2(4) = *d*

2√ 2 = *d*

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

√ 2(32) = *d*

√ 2(9) = *d*

3√ 2 = *d*

You can now see a pattern developing. The diagonal of a square is equal to any side of the square times √ 2 .

See whether you can solve the flashcards below.

TAKE YOUR TIME!

- What is the diagonal of the square above?
- What is the area of the square above?4
- What is the perimeter of the square above?8
- What's the area of the isoceles triangle?2

- What is the diagonal of the square above?
- What is the area of the square above?25
- What is the perimeter of the square above?20
- What's the area of the isoceles triangle?12
^{1}/_{2}

- What is the diagonal of the square above?
- What is the area of the square above?256
- What is the perimeter of the square above?64
- What's the area of the isoceles triangle?128

Assume that the circle touches all four sides of the square in the diagram above.

- What is the diameter of the circle above?6
- What is the area of the square above?36
- What is the perimeter of the square above?24
- What's the circumference of the circle?
- What's the area of the circle?