Pythagorean Theorem As Sweeping Area

The Pythagorean Theorem is often taken as a fact aboutright 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 namedLine AandLine B). We can spin them to create area:

Ok, fun enough. Where's the mystery?

Well, what happens if wecombinethe line segments before spinning them?

Whoa. The area swept out seems to change. Should simplymoving这些线，不是延长它们，改变了面积?

Running The Numbers

从上图可以看出，这个区域确实在增长。让我们来解决细节问题。

例如，假设$a = 6$， $b = 8$。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.

让我们找一个聪明的解决办法。Intuitively, the question is:How can Line A's lengthnothelp 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, thedot productis zero: $a \cdot b = 0$).

Ah -- that's the meaning of the Pythagorean Theorem.当线段垂直时，无论线段合并还是分离，扫过的区域都是相同的。

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.

在我们的例子中，我们有圆A = $36\pi$，圆B = $64 \pi$， $c = \sqrt{36 + 64} = 10$。戒指宽度为$10 - 6 = 4$。

Summary

毕达哥拉斯定理不仅仅是关于三角形。当分量垂直时，它们所形成的面积与它们的排列方式无关。

Appendix: Assorted Thoughts

• TheLaw of Cosinesexplicitly 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 frommultiple 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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