Volume: Prisms, Cylinders, Pyramids, Cones, Spheres
Volume is base area times height — and pointy solids hold exactly one-third of their bounding prism.
1 Key ideas
Three rules cover every solid: V = Bh for straight sides, ⅓·Bh for pointy ones, and the sphere’s own formula.
Prisms & cylinders — V = Bh
V = Bh — base area × height. Rect prism: V = lwh. Cylinder: V = πr²h.
Pyramids & cones — ⅓·Bh
V = ⅓Bh — a pointy solid holds one-third of the prism (or cylinder) around it.
Cone: V = ⅓πr²h.
Sphere
V = ⁴⁄₃πr³ — the only solid with its own formula here.
Use π = 3.14, round to 2 decimals.
Never forget the ⅓: pyramids and cones are pointy — multiply by one-third.
A cone does NOT have the same volume as its cylinder.
Square the radius first. In πr²h, compute r² before multiplying by h.
Writing out r × r stops the most common slip.
Quick check 1 — Cube with edge 3 cm. Volume?
Quick check 2 — Rectangular prism 2 × 3 × 4 cm. Volume?
Quick check 3 — Cylinder r = 2 cm, h = 6 cm. Volume? (π = 3.14)
Quick check 4 — Square pyramid: base 6 cm, height 5 cm. Volume?
2 Worked examples
The ⅓ factor in action, plus the sphere.
Example 1 — cone
Cone: radius 3 cm, height 4 cm (π = 3.14).
- Base area: π × 3² = 3.14 × 9 = 28.26.
- Pointy → multiply by ⅓: ⅓ × 28.26 × 4 = 37.68 cm³.
Example 2 — sphere
Sphere: radius 3 cm (π = 3.14).
- V = ⁴⁄₃ × 3.14 × 3³ = ⁴⁄₃ × 3.14 × 27.
- = 4 × 3.14 × 9 = 113.04 cm³.
Example 3 — triangular prism
Right triangular prism: legs 3-4, length 6 cm.
- Base area: ½ × 3 × 4 = 6.
- V = Bh = 6 × 6 = 36 cm³.
Quick check 5 — Cone r = 3 cm, h = 4 cm. Volume? (π = 3.14)
Quick check 6 — Sphere r = 3 cm. Volume? (π = 3.14)