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The circular mil "is a unit of area, equal to the area of a circle with a diameter of one mil."

"The area in circular mils, A, of a circle with a diameter of d mils, is given by the formula:"

$A=d^2$.

I can comprehend the truth of the above formula using the more standard area formula in terms of the unit square $A=\pi r^2$, but I'm wondering how to prove it geometrically?

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Let's make up another circular area unit. Let it be the area of a circle of $48$ versts, verst being an old Russian length unit.

So, we consider

$$A=48^2\cdot \pi \ \text{versts}^2$$ to be the unit of the area measurement in the case of circles.

We can consider $A=1$ by definition.

What is the area of a circle of $d\cdot 48$ versts if we measure the area in units just introduced? Obviously

$$A_d=(d\cdot 48)^2\cdot \pi=d^2\cdot (48^2\cdot \pi)=d^2\cdot A \ \text{versts}^2=d^2 \text{versts}^2$$ because $A=48^2\cdot \pi$ is $1$ unit called $\text{verst}^2$.

So, this is not a geometrical question. This is a question of choosing a special area unit. If for somebody the verst is the natural unit then calculating the area of circles in square meters is a similar problem.

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