Similar Solids and Cavalieri's Principle — Worked Examples

A geometric model is enlarged so every corresponding length is multiplied by $1.50$. Determine the corresponding area and volume multipliers.

A prototype vessel has capacity $2.40\,\text{m}^3$. A geometrically similar vessel is built at twice every internal linear dimension. Determine its capacity.

Two similar solids have corresponding surface areas $54.0\,\text{cm}^2$ and $150\,\text{cm}^2$. Find the linear scale factor from the smaller to the larger solid.

Two similar concrete specimens have volumes $0.0640\,\text{m}^3$ and $0.216\,\text{m}^3$. A corresponding edge of the smaller specimen is $0.800\,\text{m}$. Determine the matching edge of the larger specimen.

A solid bronze model has mass $18.0\,\text{kg}$. A geometrically similar model of the same bronze is made with every dimension $0.600$ times the original. Determine its mass.

A steel model is geometrically similar to an aluminum prototype at linear scale factor $k=0.500$. Let the steel density be $7850\,\text{kg/m}^3$ and aluminum density be $2700\,\text{kg/m}^3$. Determine the steel-to-aluminum mass ratio.

A right prism and an oblique prism have identical base areas of $12.5\,\text{m}^2$ and the same perpendicular height of $7.20\,\text{m}$. The oblique prism has lateral edges $8.60\,\text{m}$ long. Compare their volumes.

A prism and a pyramid have the same base area $B=30.0\,\text{m}^2$ and the same perpendicular height $h=6.00\,\text{m}$. Show why these data alone do not satisfy Cavalieri's conclusion.

A cone is $18.0\,\text{m}$ high with base area $254\,\text{m}^2$. Determine the area of a section parallel to the base located $12.0\,\text{m}$ from the apex.

A plane parallel to the base of a pyramid is located at $40.0\%$ of the full height measured from the apex. What fraction of the full pyramid volume lies between the apex and this plane?

A family of similar solids is enlarged by a linear factor $k=2.50$. Determine the factor by which its surface-area-to-volume ratio changes.

A rectangular solid has original volume $48.0\,\text{m}^3$. Its $x$, $y$, and $z$ dimensions are independently scaled by $1.20$, $0.800$, and $1.50$, respectively. Determine the new volume and state whether the new solid is necessarily similar to the original.