# The perimeter of a rhombus is 100, and one of its diagonals is 40. Find the area of a rhombus?

Since all sides of the rhombus are equal, the length of the side of the rhombus a is equal to a quarter of its perimeter:

a = 100/4 = 25.

The diagonals of the rhombus intersect at right angles and are divided in half at the intersection point, thus we have a right-angled triangle in which the hypotenuse is the side of the rhombus, the legs are half of the diagonals of the rhombus. We find the square of half of the unknown diagonal as the difference between the squares of the side of the rhombus and half of the known diagonal:

(d ^ 2/2) ^ 2 = a ^ 2 – (d1 / 2) ^ 2;

(d ^ 2/2) ^ 2 = 25 ^ 2 – 20 ^ 2;

(d ^ 2/2) ^ 2 = 625 – 400 = 225;

d ^ 2/2 = √225 = 15;

d ^ 2 = 15 * 2 = 30.

Knowing the lengths of the diagonals of the rhombus, we find its area as half the product of the diagonals:

S = 0.5 * d1 * d2 = 0.5 * 40 * 30 = 600.