A 4-sided regular
polygon
with all sides equal, all interior angles 90° and whose location on the
coordinate plane
is determined by the
coordinates
of the four
vertices (corners).

Try this
Drag any vertex of the square below. It will remain a square and its dimensions calculated from its coordinates.
You can also drag the origin point at (0,0), or drag the square itself.

In coordinate geometry, a square is similar to an ordinary square
(See Square definition )
with the addition that its position on the
coordinate plane
is known.
Each of the four vertices (corners) have known
coordinates.
From these coordinates, various properties such as width, height etc can be found.
It has all the same properties as a familiar square, such as:

The dimensions of the square are found by calculating the distance between various corner points.
Recall that we can find the distance between any two points if we know their coordinates.
(See Distance between Two Points )
So in the figure above:

The length of each side of the square is the distance any two adjacent points (say AB, or AD)

The length of a diagonals is the distance between opposite corners, say B and D (or A,C since the diagonals are congruent).

This method will work even if the square is rotated on the plane (click on "rotated" above). But
if the sides of the square are parallel to the x and y axes,
then the calculations can be a little easier.
In the above figure uncheck the "rotated" box
and note that The side length is the difference in y-coordinates of any left and right point - for example A and B.

Example

The example below assumes you know how to calculate the distance between two points, as described in
Distance between Two Points.
In the figure above, click 'reset', 'rotated' and 'show diagonals'

The side length of the square is the distance between any two adjacent
vertices.
Let's pick B and C.
Using the formula for the distance between two points:

The length of a diagonals is the distance between any pair of opposite vertices.
In a square, the diagonal is also the length of a side times the square root of two:

In the figure at the top of the page, click on "hide details" .
Then drag the square or any of its corners to create an arbitrary square.
Calculate the width, height and the length of the diagonals.
Click 'show details' and "show diagonals" to verify your answer.

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place.
This can cause calculatioons to be slightly off.