Why Walking Across the Grass is Faster: Pythagoras Explained

The Shortcut in the Park

30m

40m

You're standing at one corner of a park. The opposite corner is where you need to be. Two options: walk the sidewalk — 30 meters east, turn, 40 meters north — or cut straight across the grass.

Intuition says the grass is faster. But how much faster? That's not a guess — it's a calculation. And the answer is exactly 20 meters saved.

I use this one more than any other formula from school. Last month I was figuring out whether a 65-inch TV would fit in a 58-inch-wide alcove. Diagonal measurement, right angle, same math. The Pythagorean theorem isn't a classroom relic — it's the universe's shortcut formula.

How Far You Actually Save by Cutting Across

Let's settle the park question with real numbers before touching the formula.

Sidewalk vs. Grass

Sidewalk: 30m + 40m = 70 meters
Grass: $\sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50$ meters
Saved: 20 meters — nearly 29% shorter

That 30-40-50 is a Pythagorean triple (a scaled-up version of the famous 3-4-5). The formula behind it:

a^2 + b^2 = c^2

Square the two short sides, add them, take the square root. That's your diagonal distance — the Euclidean distance between two points.

a & b
The two legs: The sides that form the right angle (90°). In the park, that's the east and north sidewalks.
c
The hypotenuse: The longest side, opposite the right angle. Always shorter than $a + b$ combined — that's why shortcuts work.

Builders, TV Buyers, and Game Developers All Use This

The Builder's 3-4-5 Rule

How do you check if a wall corner is exactly 90° without a protractor?

Measure 3 ft along one wall.
Measure 4 ft along the other.
Measure the diagonal between those marks.

If it's exactly 5 ft, the corner is a perfect right angle.

Will That TV Actually Fit?

TVs are sold by diagonal size. A "65-inch" TV doesn't mean 65 inches wide. You need the theorem in reverse:

\text{Width} = \sqrt{65^2 - \text{Height}^2}

For a standard 16:9 ratio, that 65-inch diagonal is about 56.7 inches wide. Tight fit for a 58-inch alcove — but it works.

Game developers use the same formula constantly. Every time a game calculates the distance between two characters on a 2D map, it's running $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ — Pythagoras in disguise. If you're curious about how that distance formula connects to slope calculations, the link is more direct than you'd think.

Frequently Asked Questions

Does the Pythagorean theorem work for all triangles?

No — only right triangles (ones with a 90° angle). For other triangles, you'd need the Law of Cosines, which is a generalized version. But any time you see a right angle — a wall corner, a screen diagonal, a coordinate grid — Pythagoras applies.

What's the 3-4-5 rule and why do builders use it?

3-4-5 is the smallest Pythagorean triple — three whole numbers where $3^2 + 4^2 = 5^2$ . Builders measure 3 ft and 4 ft along two walls; if the diagonal is exactly 5 ft, the corner is a perfect right angle. No protractor needed. Multiples work too: 6-8-10, 9-12-15, 30-40-50.

How do I find the hypotenuse of a right triangle?

Square both legs, add them together, and take the square root: $c = \sqrt{a^2 + b^2}$ . For legs of 5 and 12: $\sqrt{25 + 144} = \sqrt{169} = 13$ .

Back to that park. The sidewalk route is 70 meters. The grass is 50. You save 20 meters every single time — and now you know exactly why, not just that it "feels shorter." Next time you're eyeing a diagonal shortcut, a TV spec sheet, or a game map, the math is the same three letters: $a^2 + b^2 = c^2$ .

Solve Your Triangle

Plug in any two sides — we'll find the third and show the work. Works for finding the hypotenuse or either leg.

*Also handles reverse calculations — enter the hypotenuse and one leg to find the other.