The volume of a prism or cylinder is V = B * h, where B is the area of the base and h is the perpendicular height. This follows from Cavalieri's principle: solids with equal cross-sectional areas at every height have equal volumes. For a rectangular prism, V = lwh. For a cylinder, V = pi*r^2*h. Volume measures the space enclosed by a three-dimensional figure.
Start with unit cubes to build intuition for volume as "layers of area." Show that stacking identical cross-sections produces the volume formula. Practice with various base shapes. Introduce Cavalieri's principle as the underlying justification. Give problems requiring unit conversions.
You've already worked with the surfaces of prisms and cylinders — unfolding them into nets and computing total surface area. Volume is a different question: instead of measuring the wrapper around a solid, you're measuring the space inside it.
The core idea is stacking layers. Imagine slicing a rectangular prism into thin horizontal sheets, each identical to the base. If the base has area B and you stack layers to a height h, the total volume is B × h. This is not just a formula — it's a physical fact: volume accumulates as area stacked over height. For a rectangular prism with a 4 × 3 base, each layer contributes 12 square units; 5 layers gives 60 cubic units. The unit change from square units to cubic units reflects this: you're multiplying an area (two-dimensional) by a length (one-dimensional) to get a three-dimensional measure.
Cavalieri's principle extends this reasoning to oblique (tilted) prisms and cylinders: if two solids have the same height and the same cross-sectional area at every level, they have equal volumes. Imagine a stack of coins standing straight versus leaning sideways — the same number of coins, each the same size, contain the same total metal regardless of how they're tilted. This is why V = Bh works for all prisms and cylinders, not just upright ones, as long as you use the perpendicular height rather than the slant length.
For a cylinder, the base is a circle with area πr², giving V = πr²h. A common error is computing πr·h — forgetting to square the radius — which has the wrong units (length²) and is dimensionally incorrect. Notice that volume and surface area formulas look similar but differ in dimensional structure: surface area is measured in units² while volume is in units³. As you move on to pyramids and cones, you'll discover that V = (1/3)Bh — they hold exactly one-third the volume of the corresponding prism or cylinder with the same base and height, a fact that Cavalieri's principle also helps explain.