Sorry about the crappily drawn figure but you should get the

If area is Pi/4 then the radius is 1/2; thus the diameter is 1. SO, the width of the rectangle is 3. The height is PQ+1. If you are saying that PQ is 3 or whatever it is, just add 1 to it.

The area will be 3x(PQ+1)

Here is the graphical explntn

they get more difficult than this

ans: 9pi+3

I am not sure if PQ is 3Pi, since 3Pi is close to 10.

The distance between P and Q is radical three

3 + sq rt of 3.

The ETS trick is here the radius of the width are missing but you know it is 1/2.

Victor

if the area of each circle is pi/4, then pi/4 = pi r^2
and r = 1/2
so the width of the rectangle is 3
the 2 equilateral triangles in the middle have sides of length 1
so their heights are sqrt(3)/2
the height of the rectangle is
1/2 + 1/2 + sqrt(3)/2 + sqrt(3)/2
= 1 + sqrt(3)
so the area of the rectangle is 3 + 3 sqrt(3)

Seems pretty straightforward, without any need to figure out triangles and so on...

Step 1:
We know that the area of each circle is pi / 4. Therefore, we can first figure out the radius of the inscribed circles:

Area of a circle = pi*r^2

Given the above, we solve for r:

pi/4 = pi*r^2 --> r = 1/2

We can now figure out the width of the rectangle thanks to the three bottom circles (the sum of the diameters of the three circles):

2*r*3 = 2 * 1/2 * 3 --> w = 3

Step 2:
The distance from P and Q to their respective side of the rectangle is one r, i.e. 1/2. We already know that the distance from P to Q is pi*3. Therefore, the length of the rectangle is:

r + PQ + r = 1/2 + pi*3 + 1/2 = pi*3 + 1

Step 3:
We now know the width and the length of the rectangle and can figure out the area of the rectangle:

length * width = (pi*3 + 1) * 3 = 9pi + 3