# Areas of figures with imaginary components

$n,a,b$ are positive integers.

Say, I have a square. It has $S = ni$. We know, $A = S²$. Therefore, $A = n²i² = -n²$. Generalizing to a rectangle, we get $A = abi² = -ab$.

Now, a circle. We know, $A = πr²$. Let $r = ni$. Therefore, $A = πi²n² = -πn²$. Generalizing to an ellipse, we get $A = πabi² = -πab$.

Now, an equilateral triangle. It has $S = ni$. We know, $A = √3×S²/4$. Therefore, $A = √3n²i²/4 = -√3n²/4$.

The problem here, is that despite the side/radius being positive, the areas work out to be negative. This seems to be a contradiction. How do I resolve this?

Area = $|ni × ni|$
= $|-n^2|$
= $n^2$