The Pythagorean Theorem is often taken as a fact about right triangles.

Let's try a broader interpretation: **The Pythagorean Theorem explains how 2D area can be combined.**

Here's what I mean. Suppose we have two lines lying around (the creatively named *Line A* and *Line B*). We can spin them to create area:

Ok, fun enough. Where's the mystery?

Well, what happens if we *combine* the line segments before spinning them?

Whoa. The area swept out seems to change. Should simply *moving* the lines, not lengthening them, change the area?

## Running The Numbers

Eyeballing the diagram above, it sure seems like the area grew. Let's work out the specifics.

As an example, suppose $a = 6$ and $b = 8$. When they're swept into circles ($\text{area} = \pi r^2$) we get:

For a total of $36\pi + 64\pi = 100\pi$.

The combined segment has length $c = a + b = 14$, and when we spin it we get:

Uh oh. That's way more area than before.

## The Problem

What happened? Well, Circle A didn't change. But Circle B is much less than Ring B (just look at it!).

The issue: When Line B spins on its own, it can only reach 8 units out as it sweeps. When we attach Line B to Line A, it reaches out 6 + 8 = 14 units. Now the circular sweep covers more area, meaning Circle B is smaller than Ring B.

Mathematically, here's what happened.

Ignore $\pi$ for a moment since it's a common term. When expanding $c^2 = (a + b)^2 = a^2 + 2ab + b^2$, there's a new $2ab$ term that has to go somewhere. Because Circle A doesn't change, this extra area must appear in Ring B.

## Making Things Line Up

It... sort of makes sense that the area changes, but I don't like it. Just moving things around shouldn't have this effect! Can the area ever be the same?

Sure, if we remove the $2ab$ term. The easy fix is to set $a=0$, but that's cheating and you know it.

Let's find a clever solution. Intuitively, the question is: **How can Line A's length not help Line B as it spins?**

Tilt it! As we rotate Line B, there's less benefit from Line A's length. Ladders are useless when lying on the floor, right?

When we go Full Perpendicular™, the $2ab$ term disappears and Circle B = Ring B. (In vector terms, the dot product is zero: $a \cdot b = 0$).

Ah -- that's the meaning of the Pythagorean Theorem. **When line segments are perpendicular, the same area is swept whether the lines are combined or separated.**

## Checking The Math

It's not a bad idea to make sure the numbers line up.

Since the segments are now perpendicular, we know $c^2 = a^2 + b^2$, so:

Now we can calculate:

Tada! The Ring and Circle sweep the same area.

In our example, we have Circle A = $36\pi$, Circle B = $64 \pi$, $c = \sqrt{36 + 64} = 10$. The ring width is $10 - 6 = 4$.

## Summary

The Pythagorean Theorem is about more than triangles. When components are perpendicular, the area they make is independent of how they are arranged.

## Appendix: Assorted Thoughts

- The Law of Cosines explicitly shows the $2ab$ term which assumed to be zero in the Pythagorean Theorem. The area of Ring B can even be "negative" if we tilt Line B to point inside.
- We can combine area from multiple dimensions ($x^2 + y^2 + z^2 + ...$). As long as they are mutually perpendicular, the area swept by each dimension is the area swept by the total.
- The Pythagorean Theorem is a relationship in the 2D area domain ($c^2 = a^2 + b^2$). We start here and convert this to a relationship in the 1D domain ($c = \sqrt{a^2 + b^2}$). The conversion happens so often we forget where it began.
- More on sweeping area: https://www.cut-the-knot.org/Curriculum/Geometry/PythFromRing.shtml

Happy math.

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