Circles in rectangle

• Extend the diagram: The length of the rectangle is $(24 \times 2) + (24 \times 2)$ . The breadth of the rectangle can also be broken down into 3 parts similarly, see below.

• Find $x$ : Using Pythagorean theorem,

$x^2 + 24^2 = (24 + 16)^2 \quad \implies \quad x = 32$

• Blue area: is obtained by subtracting area of circles from the area of rectangle

$\begin{aligned} \text{Blue area} &= \text{Area of Rectangle} - \text{Sum of areas of circle} \\ \\ &= (96 \times 72) - (\pi \times 24^2 + \pi \times 24^2 + \pi \times 16^2) \\ \\ &= 6912 - \pi (1408) \\ \\ \text{Blue area} &\approx \boxed{2489} \end{aligned}$

As the radius of the two circles is equal=24 cm , hence the length of the rectangle would be equal to 48+48=96 cm

now for finding the breadth of the rectangle join the centers of the circles. ..It would form an isosceles triangle with respective sides 40cm ,40 cm ,48 cm

now with the help of herons formula find the area of the triangle

and equate it with 1/2bh to find the height ...The height would be 32 cm

now the breadth of the rectangle would be 32 cm (h) +16cm (radius of smaller circle) + 24cm (radius of bigger circle)=72 cm.

now the area of the rectangle would be 72 cm *96 cm=6912 cm^2

calculate the area of the three circles and subtract it from the area of rectangle

to find the area of the shaded region =2486.8571cm^2~2487cm^2

The Width of the Rectangle is 4(24) = 96 cm

The horizontal distance between the large and small circle centres is 24 cm and the diagonal distance is 24+16 = 40 cm.

As such, the vertical distance is $\sqrt{40^2-24^2} = 32$ cm

The Height of the Rectangle is $16 + 32 + 24 = 72$ cm

This gives the Area of the Rectangle as 96 cm * 72 cm so the

Blue Area $= 96*72 - 2*\pi(24)^2-\pi(16)^2 = 6912 - 1408\pi = 2488.6375 \rightarrow \fbox {2489}$

nice solution: