Why does a parabola curve share an equation with the area of a square?

Question

Probably, $y = x^2$ plots a parabola only given certain assumptions that structure a cartesian coordinate plane, and it does not plot a parabola in e.g. the polar coordinate plane.

Now, why exactly does a parabola share an equation with the area of a square? 'Why' here is to be understood as inquiring at the equation's suggestion of a -geometrical- correspondence between the two given certain assumptions, but only the equation suggests this and not the actual shapes. Is this completely accidental, i.e., does the geometry of a parabola have nothing to do with that of a square, or does the equation $y = x^2$ indeed suggests some sort of relationship between the two shapes?

Most of all, I want to know: can we manage to identify any geometrical correspondence between a square and a parabola due to the equation?

(The equation of a circle in cartesian coordinates similarly bothers me, but at least we can speak of some sort of relationship between pythagorean triples.)

Answer 1

If you take the graph of $y = x$, the region under the graph between $0$ and $t$ is half of a square of side length $t$, and $\int_0^t x \, \mathrm{d}x = \frac{t^2}{2}$. So some sort of answer is "because the gradient of the parabola is linear and thus carves out half a rectangle".

Answer 2

The parabola is the graph of a function which plots the square of the ordinate. That happens to be the same as the function with gives the area of a square given the side.

The name should give you a hint why.

Answer 3

y = x * x says To compute the number y, multiply the number x with itself.

If you're talking about length, that corresponds to area of a square.

If you're talking about real numbers, you get a parabola curve.

If you're talking about complex numbers or some other set, you may get something else.

In other words, it depends on how you define "number" (what set of numbers you're working with) and how you define "multiply" (presumably repetitive addition, which assumes you have a definition of "addition").