The diagram shows 3 identical circles inside a rectangle
Each circle touches two other circles and the sides of a rectangle as shown in a diagram
The radius of each circle is 28mm.
Work out the area of a rectangle
Give the correct answer to three significant figures.

Let us start finding the length of the rectangle, because it is the easiest part.

In the figure attached focus on the circles with center at the points A and C. Notice that the segment AC is parallel to the segment FG, and the length of AC is 2r, where r stands for the radius of the circles. Now, it is not difficult to see that if we extend the segment AC in both directions, until it intersects with the sides of the rectangles, the resultant line has length 4r. This last assertion is consequence of the fact that the circles are tangent with the sides of the rectangle.

Thus, the length of the segment FG is 4r=4*28mm = 112 mm =0.112m.

Now, let us find the length of the segment FJ. This is step is less simple. Notice that the length of FJ is equal to KA+LD+DM, because FJ, KA and LM are all parallel to each other. We already know that KA=r, also DM=r too. So, we  only need to find the length of LD and this can be done using the Pythagorean theorem.

Recall that the triangle ACD is equilateral and the length of its sides is 2r=56mm, because the three circles are equals.  Now, as LD is the height of ACD relative to AC, using the Pythagorean theorem AL²+LD²=AD². Then,

LD²=AD²-AL²=(56)²-(14)²=3136-196=2940

Thus, LD=54.22.

Finally, the length of FJ= 28 + 54.22+28 = 56+54.22=110.22mm=0.11022 m

So, the area of the rectangle is A= 0.112*0.11022=0.01234464 m² = 1.23*10^(-2)m².

Step-by-step explanation:

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