Volume of Prisms and Cylinders
Apply V = Bh for prisms and V = πr²h for cylinders. Solve multi-step problems involving volume in engineering, architecture, and science contexts. Address capacity vs. volume concepts.
A lesson of Geometry, in Area and Volume, about 10 minutes.
By the end of this lesson, you will be able to find the volumes of right prisms and right cylinders, interpret volume as stacking base layers, and solve applied problems involving filling or capacity.
The Big Idea
Volume is the measure of three-dimensional space enclosed by a solid, in cubic units. For prisms and cylinders, the volume formula is universal and elegant: Volume = Base Area × Height. This works because these solids can be thought of as stacking infinitely thin copies of the base, layer by layer, up to height $h$.
Prisms
$$V = Bh$$
where $B$ = area of the base and $h$ = height (perpendicular distance between bases).
This works for any right prism regardless of the base shape.
Cylinders
$$V = \pi r^2 h$$
This is the same formula $V = Bh$ where $B = \pi r^2$ (area of circular base).
Example 1 — Rectangular Prism
Box: $8 \times 5 \times 3$.
$$V = (8 \times 5)(3) = 40 \times 3 = 120 \text{ cubic units}$$
Example 2 — Triangular Prism
Triangular base: legs 3 and 4; prism height = 10.
$$B = \frac{1}{2}(3)(4) = 6$$
$$V = 6 \times 10 = 60 \text{ cubic units}$$
Example 3 — Cylinder
Radius 5, height 11.
$$V = \pi(25)(11) = 275\pi \approx 863.9 \text{ cubic units}$$
Example 4 — Finding Height from Volume
Cylinder volume = $180\pi$, radius = 6. Find height.
$$180\pi = \pi(36)h \implies h = 5$$
Example 5 — Application: Pool Volume
Rectangular pool: 30 ft × 15 ft × 5 ft deep.
$$V = 30 \times 15 \times 5 = 2250 \text{ cubic feet}$$
Since 1 cubic foot ≈ 7.48 gallons: $2250 \times 7.48 \approx 16,830$ gallons.
Common Mistakes
Mistake 1: Using slant height instead of perpendicular height
For oblique prisms/cylinders, height is the perpendicular distance between the bases — not the edge length.
Mistake 2: Confusing volume (cubic) and surface area (square) units
Volume is in cubic units (cm³, ft³). Surface area is in square units (cm², ft²).
Mistake 3: Forgetting to compute $B$ first
$V = Bh$ requires computing the base area first. Don't substitute the base dimensions without computing area — especially for non-rectangular bases.
Summary
All prisms and cylinders: $V = Bh$ (base area × height)