Composite solids and conservation of volume

Add volumes of non-overlapping pieces and subtract cavities. For external surface area, omit every interface where two pieces join. For remoulding or displacement, equate the relevant volumes only when no material is lost and the stated assumptions apply.

Example

A cylinder of radius 3 cm and height 8 cm is topped by a hemisphere of the same radius. Volume \(=\pi(3^2)(8)+\dfrac23\pi(3^3)=90\pi\text{ cm}^3\). Exposed surface area, including its bottom disk, is \(2\pi(3)(8)+2\pi(3^2)+\pi(3^2)=75\pi\text{ cm}^2\). The joined circular face is internal.

A solid sphere of radius 6 cm is melted into identical spheres of radius 2 cm without loss. Their number is \(\dfrac{\tfrac43\pi(6^3)}{\tfrac43\pi(2^3)}=27\). Surface area is not conserved: new surfaces are created when the material is divided.