Transcription
What does this perfectly straight line have in common with this crazy distorting triangle? To find out, we have to investigate the physical geometry because every student is taught that the inside angles of a triangle add up to 180 degrees. But what is 180 degrees? 180 degrees is just a straight, flat line. So how does a three-sided shape hide a flat line inside of it?
Let's freeze the triangle and color the inside corners red, green, and blue. If we want to see what they add up to, we just have to physically add them up. Watch what happens when we fold this triangle. First, we grab the top corner and we fold it down so the tip touches the bottom edge. Next, we grab the bottom left corner and we fold it inward to meet the first tip. Finally, we take the bottom right corner and we fold it in. Look at the three colored corners. They've perfectly locked together. And what shape do they form? A perfectly straight line. So we have established an absolute geometric truth: One triangle equals 180 degrees equals one straight line.
Okay, so a triangle is 180 degrees. But what about a massive, distorted four-sided shape? Most people memorize that a perfect square has 360 degrees. But they have no idea why this random kite also has 360 degrees. We don't need to memorize a new rule. We just need to find the triangles.
If we draw a single line right down the middle, we physically shatter this four-sided shape into two triangles. And if one triangle is 180 degrees, then two triangles must be 360 degrees, which is a full, perfect circle. Watch the four colored corners detach. They fly to the center and lock together like puzzle pieces. They form a perfect 360-degree circle.
Now, every student was forced to memorize this formula to find the angles of any polygon: Sum equals 180 * n - 2. But nobody asks why minus two. Let's look at a five-sided pentagon. If we pick one corner and shoot lasers to every other corner, it slices perfectly into exactly three triangles. Five sides, three triangles. What about a six-sided hexagon? An eight-sided octagon? Do you see the pattern? No matter how many sides a shape has, it's always made of exactly two fewer triangles than it has sides. N minus 2 is just the physical number of triangles hiding inside the shape.
But what if a shape has an infinite number of sides? Let's crank the counter up: 10 sides, 20 sides, 50 sides, 1,000 sides. As the sides multiply, the shape smooths out until it becomes a perfect circle. A circle is just an infinite polygon holding an infinite number of triangles inside it.