Lune of Hippocrates: 2 Curved Moons = 1 Triangle

Step by step

1. Two pink moons = the triangle? No π needed!
   These two pink moons, added together, have exactly the same area as the triangle. Curved shapes, and no pi needed!
2. Square corner · sides 6, 8, 10 · a half circle on each side
   The triangle has a square corner, and sides six, eight, and ten. Each side has a half circle on it. The moons are the parts of the small half circles outside the big one.
3. Way 1: each half circle's area · first, the one on side 6
   Way one: find each half circle's area. Start with the one on the side of six.
4. Radius = 6 ÷ 2 = 3 · whole circle: π × 3² = 9π
   Its radius is half of six: three. A whole circle with radius three would be pi times three squared: nine pi.
5. Half circle: 9π ÷ 2 = 4.5π
   Our shape is only half of that circle. So it's nine pi divided by two: four and a half pi.
6. Side 8: r = 4 → 16π ÷ 2 = 8π · side 10: r = 5 → 25π ÷ 2 = 12.5π
   The same way, the half circle on eight has radius four: sixteen pi, halved, is eight pi. The big one on ten has radius five: twenty-five pi, halved, is twelve and a half pi.
7. 4.5π + 8π = 12.5π → small + small = big
   Now add the two small ones: four and a half pi plus eight pi is twelve and a half pi. Exactly the big one!
8. Small half circles = moons + 2 slivers
   Now look at the pieces. The small half circles are the moons plus two thin purple slivers.
9. Big half circle = triangle + the same 2 slivers
   The big half circle is the triangle plus the same two slivers.
10. Take away the slivers → moons = triangle
    Take the same slivers away from both. What's left must be equal: the two moons equal the triangle!
11. Triangle = 6 × 8 ÷ 2 = 24 → moons = 24
    The triangle is six times eight, divided by two: twenty-four. So the moons are twenty-four too.
12. Hippocrates of Chios (Greece, ~440 BC): the first curved area found exactly
    About twenty-four hundred years ago, Hippocrates of Chios in Greece found the area of a moon like this. It was the first curved shape anyone measured exactly.
13. Right triangle: the two moons = the triangle
    So on any right triangle, the two moons together always equal the triangle.
14. On a test: half circle = π(d ÷ 2)² ÷ 2 = πd² ÷ 8 → small + small = big → moons = triangle = 24
    On a test, write the proof like this. A half circle is pi times half the width, squared, divided by two. Half the width, squared, is the width squared over four. Halve it again: over eight. Add the two small half circles: six squared plus eight squared is ten squared, so together they equal the big one. Both pictures hold the same slivers. Take them away: the moons equal the triangle, and the triangle is twenty-four.