Question

Calculate packing fraction in the following case to two decimal places. All circles are of equal radius.

Multiply the answer by 100 and enter the value.

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Answer 1

It is quite similar to the previous question, just that you will have to join the three circles along the diagonal to get the relationship between r and a.

We cannot use the circles along the side of the square since they are not touching each other.

So, diagonal of square = r + 2r + r = 4r

$\sqrt{2a}$ = 4r

So, P.F = $\frac{Areaoccupiedbycirclesinsidesquareunitcell}{Areaofthesquareunitcell}$

But how to calculate area occupied by circles easily?

Will it be 2 ${}^{\ast }$ Area of one circle?

Yes, because the circles at the corners contribute $\frac{1}{4^{th}}$ of their area to the square.

Since there are 4, we get 4 ${}^{\ast }\frac{1}{4}$ = 1

And then there is one circle inside, so 1 + 1 = 2

Now, P.F = $2{}^{\ast }\frac{\left(pi{}^{\ast }{r}^2\right)}{a^2}$

= $2{}^{\ast }\frac{\left(pi{}^{\ast }{r}^2\right)}{\left(2r^2\right)}$

= $\frac{pi}{4}=0.785$

Notice that the P.F is same as that calculated in the previous question. What can you infer from that?