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Why is the volume of a ball 4/3 πr³? Where does the four thirds come from? Push a ball into the tin it fits exactly, and it takes up two thirds of it. The tin holds 2πr³, so the ball is ⅔ × 2πr³ = 4/3 πr³. Why two thirds? Cut the ball in half and slice the dome at any height: by Pythagoras, every slice is a cylinder's slice minus a cone's slice. Stack them up, and the dome is πr³ − ⅓πr³ = ⅔πr³. Archimedes worked this out over 2,000 years ago, and is said to have asked for a ball in a cylinder on
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