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

We have a rhombus with diagonals of 12 cm and 16 cm, and a circle inscribed inside it. We need to find the ratio of the circle's area to the rhombus's area.

When a circle is inscribed in a rhombus, the circle touches all four sides of the rhombus. The radius of this inscribed circle is the perpendicular distance from the center to any side.

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In a rhombus, the diagonals bisect each other at right angles. This creates four right triangles.

From the diagram:

Half of diagonal 1 = 12 ÷ 2 = 6 cm

Half of diagonal 2 = 16 ÷ 2 = 8 cm

Using Pythagoras theorem to find side AB:

$AB = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}$

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We'll use the fact that the area of triangle AOB can be calculated in two different ways.

Using the two perpendicular sides:

$\text{Area of } \triangle AOB = \dfrac{1}{2} \times 6 \times 8 = 24 \text{ cm}^2$

Using base and height:

$\text{Area of } \triangle AOB = \dfrac{1}{2} \times AB \times OC$

Where OC is the perpendicular distance from center O to side AB.

Since the circle is inscribed in the rhombus, it touches all sides. The distance from the center to any side equals the radius of the inscribed circle.

Setting both area calculations equal:

$24 = \dfrac{1}{2} \times 10 \times OC$

$24 = 5 \times OC$

$OC = \dfrac{24}{5} = 4.8 \text{ cm}$

Therefore, radius = 4.8 cm

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Area of Circle:

$\text{Area} = \pi r^2 = \pi (4.8)^2 = \pi \left(\dfrac{24}{5}\right)^2 = \dfrac{576\pi}{25} \text{ cm}^2$

Area of Rhombus:

For any rhombus: Area = $\dfrac{1}{2} \times d_1 \times d_2$

$\text{Area} = \dfrac{1}{2} \times 12 \times 16 = 96 \text{ cm}^2$

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$\text{Ratio} = \dfrac{\text{Area of Circle}}{\text{Area of Rhombus}} = \dfrac{\frac{576\pi}{25}}{96}$

$= \dfrac{576\pi}{25 \times 96} = \dfrac{576\pi}{2400} = \dfrac{6\pi}{25}$

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When finding the radius of a circle inscribed in any polygon, we calculate the area of a triangle formed by the center and two adjacent vertices in two different ways. This technique works because the inscribed circle's radius is always the perpendicular distance from center to any side.

Final Answer: $\dfrac{6\pi}{25}$