A square is a four-sided shape with very particular properties. All sides are equal in length, and these sides intersect at 90°. This, it has four equal sides, and four equal vertices (90°). This means, that dissecting a square across the diagonal will also have specific implications. The 90° angle is divided exactly in half.

From this, it is easy to calculate the length of the diagonal using the geometry of a triangle. This is how:
The diagonal of a square is a line drawn from opposite vertices. A diagonal divides the square into two equal right-angled triangles.

The diagonal is the hypotenuse of each triangle. In fact, each diagonal divides the square into two congruent isosceles right triangles, with two vertices of 45°, as the vertical divides the square’s right angles in half.

Pythagorean Theorem

You can use Pythagorean Theorem to calculate the length of the diagonal of a square. This theorem shows that if you have a right triangle, the length of the hypotenuse is the square root of the sum of the square of the sides. 

The hypotenuse is the side which lies opposite of the right angle, and will always be the longest side. Knowing the length of two sides, we can always calculate the length of the third side of the triangle using this theorem as long as the triangle is a right-angled triangle.

Take a look at the diagram below. This shows a triangle drawn from dividing a square in half across the diagonal.

C - Hypotenuse   B  A



The Theorem of Pythagoras is written as follows:



For a square, all sides are equal, and in this case, A = B, therefore we can write that:



H, in this case, stands for the Hypotenuse, which is the Diagonal of the square we are trying to calculate. Thus, the diagonal is the square root of twice the square of one of the sides, as follows:





And so



The formula to determine the length of the diagonal of a square simplifies to the product of the length of one of the sides and the square root of 2.

How to use this calculator

To use this calculator, simply enter the length of the side of the square, and the calculator will calculate the length of the diagonal. Similarly, if all you have is the length of the diagonal, the calculator will provide the length of one of the square sides. The calculator allows you to select the unit of measurement.

All you need for this calculator is either the length of a side or the length of the diagonal. If you do not have these figures, use this general square calculator, entering the area or the perimeter, and the tool will provide values for the diagonal and sides.