Why Is a Triangle 180 Degrees? (And Why 360 in a Circle?)

Step by step

1. Is 180° proven… or made up?
   Is one eighty degrees proven? Or did someone just make it up?
2. Your calculator: a straight line = 200?
   Because your calculator says a straight line can be two hundred.
3. 180? 200? Which one is right?
   So which one is right?
4. Same length, different units: 1 m = 100 cm
   All of them. Think of length. One meter, or a hundred centimeters? Same length, different units.
5. A full turn ÷ 360 = 1 degree
   Angles work the same way. Split a full turn into three sixty equal parts, and each part is one degree.
6. Why 360? Nobody knows for sure (Babylon)
   Why three sixty? Nobody knows for sure. We got it from Babylonian astronomers.
7. 1793, France: right angle = 100 grad → straight line = 200
   In seventeen ninety-three, France tried a new unit: a right angle is one hundred. So a straight line is two hundred. Your calculator still has it: grad.
8. The number is a choice
   So the number is a choice.
9. Why is a straight line exactly half a turn?
   But here's the real question. Why is a straight line exactly half a turn?
10. Spin half a turn around a point on the line
    Pick a point on the line. Now spin everything half a turn around it.
11. The line lands on itself · the sides swap
    The line lands right back on itself. The two sides just swap.
12. The two sides are equal: each is…
    So the two sides are equal. Each one is...
13. …half a turn
    half a turn.
14. Euclid: angles on a straight line = 2 right angles
    Euclid never used degrees. He wrote: angles on a straight line make two right angles.
15. Walk around a triangle · turn at each corner
    Same idea for triangles. Walk around one. At each corner, you turn.
16. After 3 turns, which way are you facing?
    After three turns, which way are you facing?
17. The same way: one full turn
    The same way you started. One full turn.
18. Angle + turn = ½ turn · × 3 = 1½ turns
    At each corner, the angle and the turn make a straight line: half a turn. Three corners: one and a half turns.
19. Take away the full turn…
    Take away the full turn...
20. = ½ turn: the triangle's three angles
    half a turn. That's the triangle's three angles.
21. Euclid: a triangle's angles = 2 right angles = 180°
    Euclid again: a triangle's angles make two right angles. In degrees, that's one eighty.
22. The number is a choice. Half a turn is a fact.
    The number is a choice. Half a turn is a fact.
23. On a test: straight lines − one full turn = 180°
    On a test: at each corner, the angle plus the exterior angle is one eighty. The three exterior angles make one full turn, three sixty. So the angles add up to three times one eighty, minus three sixty. That's one eighty.
24. On a test, it looks like this: find x
    On a test, it can hide like this. Find x.