image/svg+xml
l=6"
r
a
h
r = radius of spherea = radius of holeh = height of end capSphere volume = Vs = 4/3 π r³Cylinder volume = Vc = l π a²2xEnd cap volume = 2 Ve = (2/3) π h² (3 r - h)Remaining volume = V = Vs - Vc - 2VeVc = l π a² = l π (r² - l²/4) = l π r² - l³ π / 42Ve = 2 (1/3) π h² (3 r - h) = (2/3) π (r - l/2)² (3 r - r + l/2) = (2/3) π (r² + l²/4 - 2 r l/2)(2r + l/2) = (2/3) π (r² + l²/4 - r l)(2r + l/2) = (2/3) π (2r³ + r²l/2 + 2rl²/4 + l³/8 - 2r²l - r l²/2 ) = (4/3)πr³ + (1/3)πr²l + (1/3)πrl² + (1/12)πl³ - (4/3)πr²l - (1/3)πrl² = (4/3)πr³ + (1/3)πr²l + (1/3)πrl² + (1/12)πl³ - (4/3)πr²l - (1/3)πrl² = (4/3)πr³ - πr²l + (1/12)πl³ V = Vs - Vc - 2 Ve = (4/3)πr³ - πr²l + (1/4)πl³ - (4/3)πr³ + πr²l - (1/12)πl³ = (1/4 - 1/12)πl³ = (1/6)πl³ (which, by the way is always = (4/3) π (l/2)³, i.e. the sphere's volume if r = l/2)for l=6, V = (1/6)π6³ = 36π
h = r - l/2r² = a² + (l/2)²a² = r² - l²/4
l/2