The diagonals divide a kite in four right-angled triangles.

One of the two diagonals divide a kite in two isosceles triangles.

(These are almost direct and immediate consequences of a kite definition).

Let us consider that diagonal. Let  be its length.

The area of one isosceles triangle is , where  is its height.

The area of the second isosceles triangle is , where  is the height of the second isosceles triangle.

The area of the kite is the sum of these expression, and the sum is equal to ,

where  is the second diagonal's length.

So, the final formula is S = , where the diagonal lengths participate.