Geometry · Foundation · 11 min
Volumes of solids
A solid’s volume is built from the areas of its thin cross sections.
WHAT HAPPENS IF…Why is a cone one third of its matching cylinder?
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A cylinder has constant section area πR². In a cone of height H, the radius shrinks linearly as R(1−z/H), and integrating its square gives πR²H/3. A sphere centered at height R has section area π[R²−(z−R)²], which integrates to 4πR³/3.
A detail that matters
Equal heights alone do not imply equal volumes; compare all cross-sectional areas.
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