# Diagonals of a square

A square has two diagonals, which are line segments linking opposite vertices (corners) of the square.
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:

## Calculator

 ENTER ANY ONE VALUE Side clear Perimeter clear Area clear Diagonal clear

Use the calculator above 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.