Volume of a bead

[Albert Chan]

Math trivia: a bead placed on the table (hole to the sky), volume = volume of sphere with diameter = height
(You can think of a sphere is a bead with no hole, ... Neat)

Here is a prove using messy integrals

Simpson's rule for bead volume is simpler:
let R = radius of sphere, r = radius of the hole, height of bead = 2 h, thus h² = R²- r²

Half-Bead volume = h/6 * (top_area + 4 * middle_area + bottom_area)

Bead volume = 2 Half-Bead
= h/3 (0 + 4 Pi ( (√(R²- (h/2)²))² - r² ) + Pi (R²- r²))
= Pi/3 h (4 R² - h² - 4 r² + h²)
= 4/3 Pi h^3

Here is a beautiful prove with no math, just visualization