Volume of Pyramids & Cones
A pyramid or cone has exactly one-third the volume of the corresponding prism or cylinder: .
Introduction
Past Knowledge
Prism/cylinder volume (12.3.1). Cavalieri's principle.
Today's Goal
Apply V = ⅓Bh for pyramids and cones.
Future Success
Sphere volume (12.3.3), similar solids (12.3.4).
Key Concepts
Volume Formulas
| Shape | Formula |
|---|---|
| Any Pyramid | |
| Cone |
The ⅓ Factor
Why ⅓? A cube can be decomposed into exactly 3 congruent pyramids with the cube's face as the base and the opposite vertex as the apex.
Since 3 pyramids = 1 cube, each pyramid = ⅓ of the cube. By Cavalieri's principle, this extends to all pyramids and cones.
∎ Pyramid = ⅓ of its enclosing prism.
Worked Examples
Square Pyramid
Base s = 6, h = 10.
V = 120 cubic units
Cone
r = 5, h = 12.
V = 100π ≈ 314.2
Find Height from Volume
Cone V = 48π, r = 4. Find h.
→
h = 9
Common Pitfalls
Forgetting the ⅓
Prism = Bh. Pyramid = ⅓Bh. The ⅓ is the most commonly forgotten factor.
Using Slant Height
Volume uses perpendicular height h, not slant height ℓ. SA uses ℓ; volume uses h.
Real-Life Applications
Sand/Grain Piles
A pile of sand naturally forms a cone shape. V = ⅓πr²h estimates the volume of material.
Ice Cream Cones
How much ice cream fits inside the cone? V = ⅓πr²h tells you exactly.
Practice Quiz
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