1
00:00:00,000 --> 00:00:03,896
There's a single equation, written down twenty-five centuries ago,

2
00:00:03,996 --> 00:00:06,500
that still shapes how we measure the world.

3
00:00:06,600 --> 00:00:10,412
Start with any right triangle. Three sides, three corners, and

4
00:00:10,512 --> 00:00:14,072
one angle that's exactly ninety degrees — the small square

5
00:00:14,172 --> 00:00:17,228
corner that makes everything else fall into place.

6
00:00:17,333 --> 00:00:21,689
Call the two shorter sides a and b. The longest side, lying opposite the

7
00:00:21,789 --> 00:00:25,465
right angle, has its own name. We call it the hypotenuse — c.

8
00:00:25,566 --> 00:00:29,333
Pythagoras's claim is striking in its simplicity. The squares

9
00:00:29,433 --> 00:00:32,566
built on the two short sides, added together, equal

10
00:00:32,666 --> 00:00:34,658
the square built on the long one.

11
00:00:34,766 --> 00:00:38,465
Build a literal square on each side. One with area a-squared.

12
00:00:38,565 --> 00:00:42,326
One with area b-squared. One with area c-squared. The claim is

13
00:00:42,426 --> 00:00:46,498
that the two small squares, combined, fill the largest one exactly.

14
00:00:46,600 --> 00:00:50,054
It sounds too clean to be true. So let's prove it without

15
00:00:50,154 --> 00:00:53,796
any algebra — just by moving shapes around inside two boxes.

16
00:00:53,883 --> 00:00:57,440
Take two identical squares, each with side length a plus b.

17
00:00:57,540 --> 00:01:00,478
They cover the same area — that part isn't up for

18
00:01:00,578 --> 00:01:03,887
debate. We're going to fill them in two different ways.

19
00:01:04,000 --> 00:01:07,697
In the first square, drop four copies of our triangle into the

20
00:01:07,797 --> 00:01:11,494
corners. What's left over is two perfect squares — one of side

21
00:01:11,594 --> 00:01:15,108
a, one of side b. Total leftover: a-squared plus b-squared.

22
00:01:15,216 --> 00:01:18,254
In the second, place the same four triangles — this

23
00:01:18,354 --> 00:01:21,884
time rotated, so their hypotenuses meet at the center. What

24
00:01:21,984 --> 00:01:25,268
remains is a single tilted square, with side exactly c.

25
00:01:25,383 --> 00:01:29,053
Both big squares started with the same area. Both swallowed

26
00:01:29,153 --> 00:01:32,503
up the same four triangles. So whatever's left over in

27
00:01:32,603 --> 00:01:34,931
each one must also have the same area.

28
00:01:35,033 --> 00:01:39,258
Which gives us the equation. The two leftover squares on one side. The

29
00:01:39,358 --> 00:01:43,213
one tilted square on the other. Equal in area, by pure geometry.

30
00:01:43,300 --> 00:01:47,581
a-squared plus b-squared equals c-squared. Not a numerical coincidence

31
00:01:47,681 --> 00:01:51,024
— a geometric necessity, written into the shape itself.

32
00:01:51,133 --> 00:01:54,038
Try it with any right triangle you like. Any side

33
00:01:54,138 --> 00:01:58,209
lengths, any orientation. The relationship holds, every single time.

34
00:01:58,316 --> 00:02:02,429
This is the Pythagorean theorem. Two and a half thousand years old,

35
00:02:02,529 --> 00:02:06,328
and still the cleanest, most stubborn fact in all of geometry.