In a polygon any line joining two non-consecutive vertices is called a diagonal. The distance between any pair of opposite vertices gives us the length of a diagonal. In the figure below, AC and BD are the diagonals of the square ABCD.

In a square, the length of each diagonal is given by s$\sqrt{2}$, where s is the length of any side.

**Properties of diagonals of a square:**

1. The diagonals of a square are perpendicular bisector of each other.

2. The diagonals of a square bisect its angles at the vertex.

3. The lengths of the two diagonals of a square are equal.

4. Each diagonal divides the square into two congruent isosceles right triangles. And so they have the same area, which is half the area of the square.**Example:** The side length of square is 12 cms. What is the length of each diagonal of the square?

Solution: The side length of the square = s = 12cms.

The length of a diagonal of the square, d = s$\sqrt{2}$ =(12)$\sqrt{2}$ = 12 $\times$ 1.414 = 16.97 cms.

Area of a square using diagonal: If the length of a diagonal is known, the area is given by, area =

$\frac{d^2}{2}$area =

$\frac{d^2}{2}$, where d is the length of either diagonal.

**Example:** Find the area and perimeter of a square with diagonal of 20 meters.

Solution: The length of the diagonal, d = 20 meters.

The area of the square =

$\frac{d^2}{2}$ =

$\frac{20^2}{2}$ = 200 meters$^2$.

The side length of the square, s = $\sqrt{area}$ = $\sqrt{200}$ = 14.14 meters.

The perimeter of the square = 4s = 4(14.14) = 56.56 meters.

The area of the given square is 200 meters$^2$ and the perimeter is 56.56 meters.