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