

Area and Perimeter of a square(Coordinate Geometry)
The area and perimeter of a square can be found given the
coordinates
of its
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 the square itself to move it.
Area
The area of a square is calculated in the usual way once the length of a side is found.
See Square definition (coordinate geometry) to see how the side length is found.
Once the side length is known the area is found by multiplying the side length by itself in the usual way.
The formula for the area is:
where s is the length of any side (they are all the same).
The "diagonals" method to find area
If you know the length of a diagonal, the area is given by:
where
d is the length of either diagonal
The length of a diagonal can be found by using the the methods described in
Distance between two points to find the distance between say A and C in the figure above.
Perimeter
A square has four sides which are all the same length.
The perimeter of a square (the total distance around the edge)
is therefore the four times the length of any side.
See Square definition (coordinate geometry) to see how the side length is calculated.
The formula for the perimeter is
where s is the length of any side (they are all the same).
Example
The example below assumes you know how to calculate the side length of the square, as described in
Square (Coordinate Geometry).
In the figure above, click 'reset'.
 The side length of the square is the distance between the points A and B. (Or any two adjacent vertices).
Here, this is 22.
 Area is the side length times itself, or 22 x 22 = 484
 Perimeter is four times the side length or 4 x 22 = 88
Things to try
 Click on "hide details" and "rotated" then drag the vertices of the square around to create an arbitrary size.
From the coordinates of the corner points, calculate the side length, then the area and perimeter of the square.
Then click on "show details" to check your result. (The results shown above are rounded off to one decimal place for clarity)

Click "reset". Create a square that has perimeter of approximately 40, note the area.

Create a square that has perimeter of twice that, or 80, and note the area.
Notice how the area increases more rapidly than the perimeter. The perimeter merely doubles, but the area increases by four times.
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.
For more see
Teaching Notes
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other Coordinate Geometry topics
(C) 2011 Copyright Math Open Reference. All rights reserved

