Circle in Rectangle, what is unshaded Area!!

Lets zoom in on one circle and its surrounding square to calculate the ratio.

Let the diameter of the circle be $d$

Area of circle $\pi { (\frac { 1 }{ 2 } d) }^{ 2 }=\frac { \pi }{ 4 } { d }^{ 2 } \approx 0.785 { d }^{ 2 }$

As you can see, the circle (unshaded) is approximately 0.785 of the square. Hence, the shaded area is $1-0.785=0.215$ of the square.

The ratio is hence $\frac { 0.215 }{ 0.785 } \approx 0.274$

so, let us assume that the radius of each circle is 1. thus, each circle has an area of 3.14 square units. now, let us find the area of the rectangle which has sides of 6 units long and 8 units wide giving us an area of 48 square units. therefore the area of the shaded region is 10.32. now, 10.32/37.68 = 0.27...

Let 'r' be the radius of one circle.

Then:-

Sides of the rectangle are 8r and 6r and its area is 48r^2.

Area of unshaded area is 12 pi r^2.

Area of shaded area is 48r^2-12 pi r^2.

Ratio of shaded area to unshaded area is (48r^2-12 pi r^2)/ (12 pi r^2)

Solving this, you get 0.273

Let $r$ be the radius of one of the circles.

$A_{circle}=\pi{}\cdot{}r^2$

The area of the surrounding square is

$A_{square}=(2r)^2=4r^2$

The shaded area is the difference between the area of the square and the area of the circle.

$A_{shaded}=4r^2-\pi{}\cdot{}r^2=(4-\pi{})r^2$

The relation between the shaded area and the unshaded area is

$\frac{A_{shaded}}{A_{circle}}=\frac{(4-\pi{})r^2}{\pi{}\cdot{}r^2}=\frac{4-\pi{}}{\pi{}}$

Which is, approximately, equal to

0.2732395447351626861510701069801148962756771659236515

Out of 12 unshaded circles,let us take only one and consider thjs ciscle( unshased is inscribed in the square).So subtracting area of circle from area of square( consider 2cm one side and radius of circle one cm),we get area of shaded portion as 0.86.for ratio we divide 0.86/3.14 ( with radius 1 area of circle will be 3.14X1 square)that is 0.27.Since there are 12 such circles but ratio of shaded to unshaded will remain the same that is 0.27 Ans K.K.GARG,India

Let the Radius of a circle be 1(unit).

The area of a circle hence becomes pi.

According to what I consider the sides of the rectangle becomes 6 and 8 respectively. The area of unshaded region comes out to be 12pi=37.69911184.Hence the area of unshaded region =Area of Rectangle - Area of the unshaded region

=48-37.69911184=10.30088816.

Now the ratio=10.30088816/37.69911184=0.273239544