A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

The area of the rhombus is half the product of its diagonals, giving 0.5 times 12 times 16 for 96 square centimeters. The side of the rhombus is the square root of the sum of the squares of half the diagonals, which is the square root of 6 squared plus 8 squared for 10 centimeters. Since a rhombus is tangential, its inradius equals the area divided by the semi-perimeter, so the inradius is 96 divided by 20 for 4.8 centimeters. The area of the inscribed circle is pi times 4.8 squared, yielding 23.04 pi, and dividing this by the rhombus area gives the ratio 6 pi to 25.

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