

Try this
Drag any vertex of the square below. It will remain a square and the length of the diagonal will be calculated.
A square has two diagonals. Each one is a
line segment
drawn between the opposite
vertices (corners) of the square.
The diagonals have the following properties:
 The two diagonals are
congruent (same length).
In the figure above, click 'show both diagonals',
then drag the orange dot at any vertex of the square and convince yourself this is so.
 Each diagonal
bisects
the other. In other words, the point where the diagonals
intersect (cross),
divides each diagonal into two equal parts
 Each diagonal divides the square into two
congruent
isosceles
right triangles.
Because the triangles are congruent, they have the same
area, and each triangle has half the area of the square.
Length of the diagonal
In the figure above, click 'reset'. As you can see, a diagonal of a square divides it into two
right triangles,
BCD and DAB.
The diagonal of the square is the
hypotenuse
of these triangles.
We can use
Pythagoras' Theorem
to find the length of the diagonal if we know the side length of the square.
As a formula:

where:
s is the length of any side

which simplifies to:

where:
s is the length of any side


Calculator
Use the calculator on the right to calculate the properties of a square.
Enter any one value and the other three will be calculated. For example, enter the side length and the diagonal will be calculated.
Similarly, if you enter the area, the side length needed to get that area will be calculated.
Coordinate Geometry
If you know the
coordinates
of the
vertices
of a square, you can calculate all the other properties, including the diagonal lengths.
For more on this, see
Square (Coordinate geometry)
Things to try
In the figure at the top of the page, click on 'reset' and 'hide details'.
Then drag any corner to create an arbitrary square.
Calculate the length of the diagonals.
Click 'show details' to verify your answer.
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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