Square Diagonal

The square belongs to the rank of regular polygons, meaning it is an equilateral quadrilateral. Being a synthesis of a rhombus and a rectangle, each of which, in turn, is a derivative figure from a parallelogram, the square combines all the properties of the above figures.

How this helps to find the square diagonal? Let's consider its two main properties:
- All sides of the square are equal (from the rhombus)
- All angles of the square are right, i.e., equal to 90 degrees (from the rectangle)

If you draw a diagonal of the square, it forms with its sides not just a right triangle (as in a rectangle), but an isosceles right triangle, which, according to the Pythagorean theorem, will connect only two parameters - the square diagonal and its side. The sides of the square will be the legs for the triangle, and the diagonal will be the hypotenuse.

To derive the formula for the diagonal from this identity, you need to place the doubled square of the side under the square root, and since the side of the square is also squared, it can be immediately taken out of the root. As a result, the formula for the square diagonal through the side will look like the side of the square multiplied by the square root of two:

This formula is applicable in all cases where it is necessary to find the square diagonal. At the same time, the task may not give the square itself but the shape of the square as an axial section of a cylinder, for example, then the length of the square diagonal is equal to the diagonal of the section.

It should also be taken into account that the intersection point of the diagonals divides them into two equal parts (property of the parallelogram), respectively, each segment obtained as a result of the intersection of diagonals will be equal to half the square diagonal.

Formulas for the square diagonal through area, perimeter