The pythagorean position is often taken as a fact about the right triangles.

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

Here is what I mean. Suppose we have two lines lying down (the creative name * Line A * and * Line B *). We can spin them to create area:

Okay, funny enough. Where is the mystery?

Well, what if we * combine the line segments before they spin them? *

Whoa. The area swept out seems to change. Should only * move the * lines, not extend them, change area?

## Running the numbers

The above eyeballing chart seems to be growing. Let's work out the details.

As an example, assume $ a = 6 $ and $ b = 8 $. When swept in circles ($ text {area} in 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 we get:

Uh oh. It is much more than before.

## The problem

What happened? Well, Circle A did not change. But Circle B is much smaller than Ring B (just look at it!).

The problem: When Line B spins itself, "it just knocks out" 8 units. When we attach line B to line A, it reaches 6 + 8 = 14 units. Now, the circular sweep covers more area, which means that Circle B is smaller than Ring B.

Mathematically here is what happened. Underbrace {b ^ 2} _ {text {Circle B}}} "alt =" displaystyle {underbrace {[a + b] ^ 2} _ {text {Circle C}} = underbrace {a ^ 2} _ {text {Circle A}} + subbrace {2ab + b ^ 2} _ {text {Ring B}}> underbrace {a ^ 2} _ {text {Circle A} } + underbrace {b ^ 2} _ {text {Circle B}}} "align =" absmiddle "class =" tex “/>

Ignore $ in $ for a moment because it is a common term. When you expand $ c ^ 2 = (a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2 $ there is a new $ 2ab $ term that must go somewhere. Since Circle A does not change, this extra area must be displayed in Ring B.

## Making things up

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

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

Let's find a smart solution. Intuitively, the question is: ** How can line A length not help line B when spinning? **

Tilt it! When we rotate line B, it has less benefit from line A length. Ladders are useless when lying on the floor, right?

When we go Full Perpendicular ™, $ 2ab $ term 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 the line segments are perpendicular, the same area is swept if the lines are combined or separated. **

## Checking the Math

It's not a bad idea to make sure the numbers are right. Since the segments are now perpendicular, we know $ c ^ 2 = a ^ 2 + b ^ 2 $, so:

Now we can calculate:

Tada! Call and circle sweep the same area.

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

## Summary

Don't think about the pythagorean theorem just about the area. When parts are perpendicular, the area they do is independent of how they are arranged.

## Appendix: Mixed Thoughts

- Cosin's law expressly expresses the $ 2ab $ term assumed to be zero in Pythagorean theorem. The area in 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 + … $). As long as they are mutually perpendicular, the area swept by each dimension is swept by the sum.
- The Pythagorean rate is a ratio in the 2D domain domain ($ c ^ 2 = a ^ 2 + b ^ 2 $). We start here and convert it to a ratio in the 1D domain ($ c = sqrt {a ^ 2 + b ^ 2} $). The transformation happens as often as we forget where it started.
- Additional reading in sweeping area: https://www.cut-the-knot.org/Curriculum/Geometry/PythFromRing.shtml

Happy math.

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