The Pythagorean theorem can apply to any shape, not just triangles. It can measure nearly any type of distance. And yet this 2000-year-old formula is still showing us new tricks.

Re-arranging the formula from this:

$\displaystyle{c = \sqrt{a^2 + b^2}}$

to this:

$\displaystyle{c= a \cdot \sqrt{1 + (b/a)^2}}$

helps us understand the relationship between slope (steepness) and distance. Letâ€™s take a look.

## Rescale Your Triangle

Scaling leads to new insights. Yes, $500k/year is a lot; but it really comes alive when you imagine things costing 10x less (A new laptop?$150. A new porsche? \$6000).

Rescaling formulas can be eye-opening as well. Letâ€™s start with our favorite 3-4-5 triangle and divide every side by 3:

What happened?

Well, we have a smaller red triangle with sides 3/3 (aka 1), 4/3 and 5/3. Weâ€™ve got a mini version of the large triangle, and the Pythagorean Theorem still holds:

$\displaystyle{1^2 + (4/3)^2 = (5/3)^2}$

## So Whyâ€™s This Special?

It doesnâ€™t seem like much, but thereâ€™s some surprising insights:

First, we can rescale any triangle to have 1 as the smallest side (divide by â€śaâ€ť). All similar triangles (i.e. those with the same ratios, like 3-4-5 and 6-8-10) will shrink into the same mini triangle.

This mini triangle has an interesting property: it only cares about the ratio b/a. The only â€śmeaningfulâ€ť numbers are 1 and (b/a), giving:

$\displaystyle{\text{mini hypotenuse} = \sqrt{1 + (b/a)^2}}$

And whatâ€™s special about b/a? Itâ€™s the slope of the hypotenuse line! Itâ€™s called the slope, the gradient, the derivative, rise over run â€” whatever the label, b/a is the rate at which the hypotenuse changes!

This makes sense. For every unit traveled along the short leg, we gain â€śslope units (b/a)â€ť on the other leg. In a 3-4-5 triangle, we go 4/3 units â€śNorthâ€ť for every 1 unit â€śEastâ€ť. And the length of our hypotenuse increases 5/3 (1.66) for every 1 unit East.

The result is pretty cool: we used the steepness of the hypotenuse (b/a) to find the distance traveled per unit East, sqrt(1 + (b/a)^2).

## An Example, Please

This is a bit weird, so letâ€™s do an example. Suppose weâ€™ve gone 5 units East and 12 units North. Whatâ€™s our distance from the starting point?

The traditional approach plugs in the Pythagorean Theorem to get c = sqrt(52 + 122) = 13. It works, but letâ€™s try our mini-triangle method:

Instead of a large triangle with sides 5 and 12, scale down by 5: we get a mini triangle with sides 5/5 (or 1) and 12/5. The â€śmini hypotenuseâ€ť is then sqrt(1 + (12/5)^2) = 2.6. This means we travel 2.6 units along the hypotenuse for every 1 unit East. Going the full 5 units East (our original triangle) is 5 * 2.6 = 13 units. Neato â€” we got the same answer both ways.

But silly me, I made a mistake. Instead of 5 units on that trajectory, I meant 6. No 7. No wait, 8. 9, for sure.

Normally, weâ€™d be furiously hammering that square root button to find the new distance. Maybe even using trigonometry to â€śmake it easierâ€ť. But not today â€” since weâ€™re on the same trajectory, we can re-use our scaling constant of 2.6:

We can find the new distance traveled with regular multiplication, with nary a square root in sight. Cool! This approach is faster for humans and computers alike â€” you wouldnâ€™t believe the crazy approaches programmers take to avoid a square root.

## Static and Dynamic Formulas

Iâ€™ve realized that our venerable Pythagorean Theorem focuses on a and b separately:

$\displaystyle{c = \sqrt{a^2 + b^2}}$

We consider a and b as separate elements, to be squared and summed. This approach is straightforward, and helps when designing bridges or making pictures of triangles. The traditional formula focuses on final values.

But the rescaled version has a new twist:

$\displaystyle{c = a \cdot \sqrt{ \left(1 + (b/a)^2 \right)} }$

Weâ€™re not that interested in the separate quantities â€” we want the ratio b/a, or the slope of the hypotenuse. This slope creates a scaling constant, sqrt(1 + (b/a)^2), that tells us how our â€śEastwardâ€ť motion translates to distance along our path. The dynamic formula focuses on rates of change.

If we have a hypothetical function f(x), we might write the dynamic Pythagorean Theorem this way:

$\displaystyle{\text{distance along path} = x \cdot \sqrt{1 + (slope)^2}}$

This concept is used in calculus to find the length of any line or curve â€” but weâ€™ll save that for another day.

The key is to realize a single formula can be re-arranged and lead to new insights. Stay curious â€” we stop learning when we think weâ€™ve â€śgot it all figured outâ€ť.

## Appendix 1: Slope vs. Distance

One point that confused me was separating the idea of slope (b/a) from distance traveled (the hypotenuse, c).

Slope is b/a, rise over run â€” how much height you get when you increase width. How â€śsteepâ€ť the hill is, so to speak. Unfortunately, the word â€śslopeâ€ť makes us think of the side of the hill â€” but slope is really about height.

Distance (the hypotenuse) is about the side of the hill â€” how far youâ€™ve walked. The â€śsteepnessâ€ť isnâ€™t that important â€” youâ€™re laying a measuring tape on the ground, which could be flat, vertical or upside-down. Does the length of a board depend on how you hold it?

But, in our man-made world, slope and distance are related because we often express locations in terms of â€śunits East (x coordinate)â€ť and not â€śunits along a pathâ€ť. So when a map says â€śgo 1 mile due Eastâ€ť and youâ€™re in front a mountain (large slope), you end up traveling a large distance (more than 1 mile). When on a flat road (zero slope), 1 mile East is simply 1 mile East. The bigger the slope, the more distance you must travel to â€śgo 1 mile Eastâ€ť.

Again, we see that the Pythagorean Theorem is not just about triangles â€” it can convert slope (steepness) into distance traveled. Happy math.