Circles Inscribed In A Rectangle

To begin solving this problem, notice that the length and height of the rectangle can be written in terms of the radius, $r$ , of the circles as such:

$Length=6r$

$Height=4r$

Now to solve for the area of the rectangle, solving for the radius is prominent. As said in the given, the combined area of all the circles is $2\pi$ . Using this information, we can solve for the area of one circle as such:

Let $x$ = area of one circle.

$\therefore 6x=2\pi$ $\Rightarrow$ $3x=\pi$ $\Rightarrow$ $\boxed{x=\frac{\pi}{3}}$

Now that we have solved for the area of one circle, we can solve for the radius by setting it equal to the formula for finding the area of a circle as such:

$\frac{\pi}{3}=\pi r^{2}$ $\Rightarrow$ $\frac {1}{3}=r^{2}$ $\Rightarrow$ $\boxed{r=\frac{1}{\sqrt{3}}}$

Now that we know the value of the radius, $r$ , we can reliably solve for the area of the rectangle as such:

$Length=6 \times \frac{1}{\sqrt{3}}$ $\Rightarrow$ $Length=\frac{6}{\sqrt{3}}$

$Height=4 \times \frac{1}{\sqrt{3}}$ $\Rightarrow$ $Height=\frac{4}{\sqrt{3}}$

$\therefore$ Area of the Rectangle = $\frac{6}{\sqrt{3}} \times {\frac{4}{\sqrt{3}}}$ $\Rightarrow$ $\frac{24}{3}$ $\Rightarrow$ $\boxed{8}$

Let the radius of each circle be $r$ $\implies diameter = 2 \times r$

Now, length of the rectangle is combination of diameters of 3 equal circles and breadth of the rectangle is the combination of diameters of 2 equal circles.

$\implies l = 3 \times 2r = 6r$

$\implies b = 2 \times 2r = 4r$

Now, area of 6 equal circles is $2π$ .

$6 \times 2πr^2 = 2π$

$\implies r^2 = \dfrac{1}{3}$

Now, area of the rectangle = $l \times b$

$\implies 6r \times 4r = 24r^2 = 24 \times \dfrac{1}{3} = 8$

Therefore, $\color{#20A900}\text{area of the rectangle is 8 .}$

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

Since the total area of all the circles combined is $2\pi$ , that means $6\pi r^2=2\pi$ $\implies r^2=\frac{1}{3}$ $\implies r=\frac{1}{\sqrt{3}}$

Because the circles are inscribed in the rectangle, that means the dimensions of the rectangle are $6r \times 4r$ :

$\frac{6}{\sqrt{3}} \times \frac{4}{\sqrt{3}} = \frac{24}{3} = \boxed{8}$