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:

- All four sides are congruent
- Opposite sides are parallel
- The diagonals bisect each other at right angles
- The diagonals are congruent

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.

**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 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

- Introduction to coordinate geometry
- The coordinate plane
- The origin of the plane
- Axis definition
- Coordinates of a point
- Distance between two points

- Introduction to Lines

in Coordinate Geometry - Line (Coordinate Geometry)
- Ray (Coordinate Geometry)
- Segment (Coordinate Geometry)
- Midpoint Theorem

- Cirumscribed rectangle (bounding box)
- Area of a triangle (formula method)
- Area of a triangle (box method)
- Centroid of a triangle
- Incenter of a triangle
- Area of a polygon
- Algorithm to find the area of a polygon
- Area of a polygon (calculator)
- Rectangle
- Square
- Trapezoid
- Parallelogram

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