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# this is so hard

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The diagram consists of a large circle and four small circles.

Each of the smaller circles has radius 1 and touches the large circle and the two small circles next on either side of it.

To 3 significant figures, find the value of the shaded area in square units?

May 22, 2020

#1
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Join the centers  of each  of  the small circles

This will create  a square  with a side  of  2  and  a diagonal  of   2√2

The area  of the square  = 2^2  =  4    (1)

Then  the  diameter  of  the  large circle  =  2 + 2√2

So  the radius  =   1 + √2

So  the area of  the  larger circle  =  pi (1 + √2)^2  =  pi  ( 3 + 2√2)    (2)

And we will  have  4 additional  white  areas  that  each  have  the area  of 3/4 of a circle with a radius of 1

So.....the total  of  the areas  =  4 (3/4) pi (1)^2  =  3 pi      (3)

So....the  grey  area  =   (2)  -  (1)  - (3)  =

pi ( 3 + 2√2)  - 4  - 3pi  =

2√2 pi  -  4   ≈

4.89  units^2   (rounded to 3 sig digits)

May 22, 2020