[bslorence]

Maybe "asking 'why'" isn't the right way to express it. Perhaps what the OP was really after here was a concrete illustration of the formula, rather than a rigorous derivation of the formula from first principles.

There are many concrete geometrical illustrations of fundamental concepts in algebra, but most people aren't even aware that they exist.

The niece's comment, oddly, both points to this and glosses over it. You don't teach a child multiplication by only forcing them to memorize times tables. You also show them stacks of coins, etc. But we do teach children algebra by only forcing them to memorize symbolic procedures. Meanwhile Book 2 of Euclid is free online, or very cheap from Dover.