The length of a segment with endpoints  and  can be found using the distance formula as follows:

This is the length of one side of the square, so the area is the square of this, or .

Since a square is a rhombus, one way to calculate the area of a square is to take half the square of the length of a diagonal. If we let  be the length of each diagonal, then 

Therefore, we want to choose the point that is  units from the origin. Using the distance formula, we see that  is such a point:

Of the other points:

:

:

:

One of your holiday gifts is wrapped in a cube-shaped box. 

If one of the edges has a length of 6 inches, what is the area of one side of the box?

We are asked to find the area of one side of a cube, in other words, the area of a square.

We can find the area of a square by squaring the length of the side.

In terms of , the area of the circle is equal to

. 

Each side of the square has length equal to the diameter, , so its area is the square of this, or

Therefore, the ratio of the area of the square to that of the circle is

Therefore, the area of the circle is multiplied by this ratio to get the area of the square:

Substituting:

To find the area of a square, we will use the following formula:

where l is the length and w is the width of the square.

Now, we know the base (or length) of the square is 9cm.  Because it is a square, all sides are equal.  Therefore, the width is also 9cm.  

Knowing this, we can substitute into the formula.  We get

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the area of the square?

To find the area of a square, simply square the side length:

So, our answer is:

Write the formula for the area of a square.

Substitute the side into the formula.

The answer is:  

Squares have all congruent sides. To find the diagonal, first recognize that you're dealing with an isoceles triangle when you draw the diagonal in the square. That means that two of the sides are congruent in the triangle. Thus, it's a special 45-45-90 triangle. In such triangles, the sides are x and the hypotenuse is . Since we know x is 4, we can plug in 4 to the expression . Thus, the answer is .

You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of , what will the length of the hypotenuse of the triangles be?

This problem is trying to distract you by thinking of triangles. What we are really asked to find here is the length of the diagonal of a square with sides of 15 inches.

Splitting a square along its diagonal yields two 45/45/90 triangles. If you know the ratios for 45/45/90 triangles, you can find the answer very quickly.

Think:

Meaning that if the two short sides are x units long, the hypotenuse will be x times the square root of two units long.

In our current case, our short sides are 15 inches long, so our hypotenuse will be 

You could also solve this with Pythagorean Theorem.

a and b are both 15 in, so we can solve.

So,our answer is 

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?

To find the diagonal of a square, we can recognize one of two things.

1) The diagonal of a square creates a right triangle, and we can use Pythagorean theorem to find our diagonal.

2) The diagonal of a square creates two 45/45/90 triangles, with side length ratios of 

Using 2), we can find that the diagonal of the square must be