Find the length of the circle inscribed in the rhombus if: the diagonals of the rhombus are 6cm. and 8 cm.

The diagonals of the rhombus intersect at right angles and are halved at the intersection. This means that the half of the diagonals and the side of the rhombus form a right-angled triangle, from which we can find the side of the rhombus:

a ^ 2 = (d1 / 2) ^ 2 + (d2 / 2) ^ 2 = 4 ^ 2 + 3 ^ 2 = 16 + 9 = 25 = 52;

a = 5 cm – rhombus side.

The area of ​​the rhombus is half the product of the diagonals:

S = d1 * d2 / 2 = 8 * 6/2 = 24 cm2.

On the other hand, the area of ​​a rhombus is equal to the product of the side length and height:

S = a * h.

Knowing the area and length of the side, we can find the height:

h = S / a = 24/5 = 4.8 cm.

It is known that the diameter of a circle inscribed in a rhombus is equal to the height of the rhombus:

d = h = 4.8 cm.

Find the length of the inscribed circle:

l = π * d = 4.8π ≈ 15.07 cm.