What volume means and its properties
Volume is the amount of space inside a solid. We measure it by counting unit cubes (1 cm × 1 cm × 1 cm is 1 cm³).
Four simple properties:
- Volume is never negative, and the unit cube has volume 1.
- Equal solids (congruent) have equal volumes. Moving or turning a solid does not change it.
- Volumes add. If a solid is cut into parts, the whole volume is the sum of the parts.
- Cavalieri's principle. If two solids have the same height and, at every level, slices of equal area, then their volumes are equal. That is why leaning the prism did not change its volume.
Parallelepiped and prism
A prism has two equal parallel bases and flat side faces. A parallelepiped is a prism whose base is a parallelogram. A cuboid and a cube are special cases.
V = B × h, where B is the area of the base and h is the height (the straight distance between the two bases, not the slanting edge).
- Cuboid: V = l × b × h.
- Cube: V = a³.
- Triangular prism: V = ½ × (triangle base) × (triangle height) × (prism length).
Pyramid: why one third?
A pyramid has a base and triangular faces meeting at one point (the apex). Its volume is
V = ⅓ × B × h
Why a third? A cube can be cut into three equal pyramids that meet at one corner. Each has the cube's base and height, so each is one third of it. Cavalieri extends this to every base shape. The slice at the middle of a pyramid is only a quarter of the base area, so a pyramid is much thinner than a prism.
Volume as an integral of cross-section area
Cut a solid into thin slices of thickness Δx. Each slice has volume (area of the slice) × Δx. Add them up. With thinner slices the answer gets better, and in the limit it becomes an integral:
V = ∫ A(x) dx from 0 to h, where A(x) is the area of the slice at height x.
For a square pyramid with base side a and height h, the slice at height x has side a(1 − x/h), so A(x) = a²(1 − x/h)². Then
V = a² ∫(1 − x/h)² dx = a²·h/3 = ⅓ × B × h. This proves the one-third rule.
Frustum of a pyramid
A frustum is what remains when a plane parallel to the base cuts off the top of a pyramid. If the bases have areas B₁ (bottom) and B₂ (top) and the height between them is h:
V = (h/3)(B₁ + B₂ + √(B₁B₂))
It comes from "big pyramid minus small pyramid". The cut-off pyramid is similar to the big one, so its sides are in the same ratio.
Try it at home
Make a paper prism and a paper pyramid with the same base and height. Fill the pyramid with rice and pour it into the prism. You will need three pours.
Key formulas and definitions
- Prism / parallelepiped: V = B × h
- Cuboid: V = l × b × h, Cube: V = a³
- Pyramid: V = ⅓ × B × h
- Frustum: V = (h/3)(B₁ + B₂ + √(B₁B₂))
- Volume from slices: V = ∫₀ʰ A(x) dx
- Scale all lengths by k: volume becomes k³ times
- Units: cm³, m³ (1 litre = 1000 cm³)
Worked examples
1. Find the volume of a cuboid 5 cm × 4 cm × 3 cm.
V = l × b × h = 5 × 4 × 3 = 60 cm³.
2. A triangular prism has a right-angled triangle base with legs 6 cm and 8 cm. Its length is 10 cm. Find the volume.
Base area = ½ × 6 × 8 = 24 cm². V = B × h = 24 × 10 = 240 cm³.
3. A square pyramid has base side 6 m and height 9 m. Find its volume.
B = 6 × 6 = 36 m². V = ⅓ × 36 × 9 = 108 m³.
4. A regular hexagonal prism has base side 4 cm and height 10 cm. Find its volume. (√3 = 1.73)
Hexagon area = (3√3/2) × side² = 2.598 × 16 ≈ 41.6 cm². V = 41.6 × 10 ≈ 416 cm³.
5. A frustum has square bases of side 6 cm (bottom) and 2 cm (top), and height 3 cm. Find its volume.
B₁ = 36, B₂ = 4, √(B₁B₂) = √144 = 12. V = (3/3)(36 + 4 + 12) = 52 cm³.
6. A square pyramid has base side 12 cm and height 9 cm. A plane parallel to the base cuts off its top 6 cm. Find the volume of the frustum left.
Whole pyramid: ⅓ × 144 × 9 = 432 cm³. The top pyramid has height 6, so its base side is 12 × 6/9 = 8 cm: ⅓ × 64 × 6 = 128 cm³. Frustum = 432 − 128 = 304 cm³. Check by the formula: h = 3, (3/3)(144 + 64 + 96) = 304 cm³.
7. A pyramid has base side 4 and height 4. Cut it into 2 equal slabs and estimate the volume using the middle width of each slab. Compare with the exact value.
Each slab is 2 thick. Middle widths: 3 and 1. Sum = 3² × 2 + 1² × 2 = 18 + 2 = 20. Exact = ⅓ × 16 × 4 = 21.33. With more slices the sum comes closer.
8. Use an integral to find the volume of a pyramid of square base a and height h.
Slice at height x has side a(1 − x/h), area a²(1 − x/h)². V = ∫₀ʰ a²(1 − x/h)² dx. Let u = 1 − x/h, dx = −h du. V = a²h ∫₀¹ u² du = a²h/3 = ⅓ B h.
Common mistakes
- Using the slanting edge instead of the vertical height h. In V = ⅓Bh or V = Bh, h is the straight distance between the bases.
- Forgetting the ⅓ for a pyramid, or using it for a prism.
- In the frustum, writing V = h × (B₁ + B₂) / 3 and leaving out √(B₁B₂).
- Mixing units: 1 m³ = 1,000,000 cm³, not 100 cm³. Convert lengths first.