Volume formulas for the four most common 3D solids — and why the ⅓ shows up exactly where it does.
The Mnemonic
Base Times Height — With One Recurring Exception
Prism: Volume = B × h, where B is the area of the base shape (any polygon) and h is the perpendicular height. Cylinder: Volume = πr²h — really just the prism formula with a circular base (B = πr²). Pyramid: Volume = ⅓Bh. Cone: Volume = ⅓πr²h — the cylinder formula with the same ⅓ applied.
Sphere: Volume = ⁴⁄₃πr³ — this one doesn't fit the base-times-height pattern at all, since a sphere has no flat base or single height; it's a genuinely separate formula built entirely from the radius.
Prisms and cylinders use full B×h; pyramids and cones use ⅓ of that; the sphere formula stands alone
💡 Memory Trick
"Volume formulas for the four most common 3D solids." Notice the pattern: pyramid is exactly ⅓ of a prism with the same base and height, and cone is exactly ⅓ of a cylinder with the same base and height. Any shape that comes to a POINT (pyramid, cone) gets the ⅓; any shape with two matching flat ends (prism, cylinder) uses the full B×h.
Why It Works
Why Pointed Solids Get Exactly One-Third
It's not a coincidence that both the pyramid and cone use exactly ⅓ of the corresponding prism/cylinder volume — this can be shown directly with a physical demonstration: three identical pyramids, each with the same square base and height, can be fit together perfectly to form one complete rectangular prism with that same base and height. Since it takes exactly three of them to fill the prism, each pyramid must be exactly ⅓ of the prism's volume.
The same relationship holds true for cones and cylinders — three cones with a matching base and height exactly fill one cylinder. This is a genuine geometric fact (provable with calculus, using integration to sum up infinitely many thin cross-sectional slices), not just a formula to memorize blindly.
Using It In A Proof
Finding the Base Area First
For prisms and pyramids (unlike cylinders and cones, which always have circular bases), the trickiest part is often finding B — the area of the specific polygon serving as the base — before the volume formula can even be applied.
1
Identify the base shape
Prisms and pyramids can have any polygon as a base — triangular, rectangular, hexagonal, etc. Identify which shape it is before finding its area.
2
Calculate B using the appropriate area formula
Use the correct area formula from Quadrilateral and Triangle Areas (or a regular polygon formula) to find B first.
3
Plug B and h into the volume formula
V = Bh for prisms, or V = ⅓Bh for pyramids — apply the ⅓ only for the pointed solids.
Full Worked Example
Finding the Volume of a Triangular Pyramid
Given: A pyramid has a triangular base with a base length of 6 and height of 4 (for the triangle itself), and the pyramid's overall height is 9. Find: the pyramid's volume.
1
Find the area of the triangular base first
B = ½ × base × height = ½ × 6 × 4 = 12.
2
Apply the pyramid volume formula
V = ⅓Bh = ⅓ × 12 × 9.
3
Calculate
V = ⅓ × 108 = 36 cubic units.
Notice two different 'heights' were involved here — the triangle's own height (4, used to find B) and the pyramid's overall height (9, used in the volume formula) — keeping these separate is essential.
🎯 Quick Worked Example
A cylinder has a radius of 5 and a height of 12. A cone has the same radius and height. Find both volumes and compare.
1
Find the cylinder's volume. V = πr²h = π(5²)(12) = π(25)(12) = 300π ≈ 942.
2
Find the cone's volume. V = ⅓πr²h = ⅓(300π) = 100π ≈ 314.
3
Compare. 314 is exactly ⅓ of 942, confirming the cone-cylinder relationship directly.
📌 Exam Application
Watch for problems that give you a solid's diameter instead of radius (same trap as with circles) — always convert to radius before squaring or cubing it in a volume formula, since using diameter directly produces a dramatically inflated (and wrong) volume.
⚠️ Most Common Volume Formulas Mistakes
Trap 1 — Forgetting the ⅓ for pointed solids: Pyramids and cones are NOT simply B×h or πr²h — always include the ⅓ for any solid that comes to a point, or the volume will be exactly 3 times too large.
Trap 2 — Confusing a 2D shape's own height with the solid's overall height: When the base is a triangle or trapezoid, that shape has its own internal height used just to calculate B — this is a completely different measurement from the solid's overall height (h) used in the volume formula itself.
✓ Quick Self-Test
1) State the volume formula for a prism. 2) State the volume formula for a pyramid, and explain why it includes ⅓. 3) A cylinder has radius 4 and height 10 — find its volume. 4) A cone has the same radius and height as that cylinder — find its volume. 5) Why doesn't the sphere volume formula fit the B×h pattern used by the other four solids?