Capacity, displacement and composite prisms

\(1\text{ cm}^3=1\text{ mL}\), \(1000\text{ cm}^3=1\text{ L}\), and \(1\text{ m}^3=1\,000\,000\text{ cm}^3=1000\text{ L}\). Cubic conversion factors are cubes of length conversion factors.

Example

A tank with internal base 40 cm by 25 cm contains water to depth 18 cm. Water volume is \(40\times25\times18=18\,000\text{ cm}^3=18\text{ L}\). A fully submerged object of volume \(1500\text{ cm}^3\) raises the water by \(\dfrac{1500}{40\times25}=1.5\text{ cm}\), provided no water overflows.

For a composite solid, add non-overlapping volumes or subtract a hole. For exposed surface area, omit joined faces and include exposed walls inside a hole. A hollow cylinder of outer radius \(R\), inner radius \(r\) and height \(h\) has material volume \(\pi(R^2-r^2)h\).

For \(R=4\text{ cm}\), \(r=3\text{ cm}\), \(h=10\text{ cm}\), the material volume is \(70\pi\text{ cm}^3\). Its exposed surface area, including both annular ends, is \(2\pi(4)(10)+2\pi(3)(10)+2\pi(16-9)=154\pi\text{ cm}^2\). A volume ratio alone does not guarantee that solid objects can physically fit inside a container.