Where Does Pi Come From? 3 Ways to See 3.14

Step by step

1. A wheel 2 ft across · how far in one turn?
   A bike wheel is two feet across. How far does it roll in one turn?
2. About 6.28 ft · the secret: π ≈ 3.14
   About six point two eight feet. The secret is a number called pi, about three point one four.
3. π = around ÷ across, for any circle · but why 3.14?
   Pi is how many times the distance across fits around a circle. Any circle! But where does three point one four come from?
4. Way 1: roll the wheel one full turn
   Way one: roll it. Mark the bottom of the wheel, and roll exactly one turn.
5. 3 × across, plus a little → π is a bit more than 3
   Now lay the distance across along the track. Three fit, plus a little bit more. So pi is a bit more than three.
6. Way 2: trap π between two shapes · a hexagon inside
   But measuring is never exact. Way two traps pi between two shapes. Put a hexagon inside the circle.
7. 6 sides = 6 radii = 3 × across → π > 3
   It's six equal triangles, so each side equals the radius. Six radii are three times across. The circle is longer, so pi is more than three.
8. Square: 4 × across → π < 4
   Now a square outside. Each side is as long as the circle is wide. That's four times across, and the circle is shorter. So pi is less than four.
9. 3 < π < 4 · double the sides → tighter
   Pi is trapped between three and four. Double the sides again and again, and the trap gets tighter.
10. Archimedes (Greece): 96 sides → 3.14 < π < 3.143
    About twenty-two hundred years ago, the Greek mathematician Archimedes used ninety-six sides. Pi was between three point one four and three point one four three.
11. Liu Hui (China, 263 AD): 3.14 → 3.1416
    In China, Liu Hui kept doubling the sides too, and found three point one four, then three point one four one six.
12. Zu Chongzhi (China, ~480 AD): 7 digits · 355/113
    Later, Zu Chongzhi pushed it to seven digits, and found the fraction three fifty-five over one thirteen. No one beat him for about nine hundred years.
13. Area: cut into slices, tip up, tip down
    Pi also gives the area. Cut a pizza into thin slices, and line them up, tip up, tip down.
14. Width = half the way around = πr · height = r → area = πr²
    It's almost a rectangle. It's half the way around wide, which is pi times r, and one radius tall. So the area is pi r times r: pi r squared.
15. C = π × d · A = π × r²
    So: around a circle is pi times the distance across, and the area is pi times r squared.
16. On a test: C = πd ≈ 3.14 × 2 = 6.28 ft
    On a test, write it like this. The diameter is two feet. Use the circumference formula. Pi times two is about six point two eight. The wheel rolls about six point two eight feet.