Cavalieri's Principle & Volume of Prisms/Cylinders
If two solids have equal cross-sectional areas at every height, they have equal volumes. For prisms and cylinders: .
Introduction
Past Knowledge
Area formulas (11.1–11.2). Cross-sections (12.1.2).
Today's Goal
Understand Cavalieri's and apply V = Bh for prisms and cylinders.
Future Success
Pyramid/cone volume (12.3.2), sphere volume (12.3.3).
Key Concepts
Volume Formulas
| Shape | Formula |
|---|---|
| Any Prism | |
| Rectangular Prism | |
| Cylinder |
Cavalieri's Principle
If two solids have the same height, and every cross-section at the same level has the same area, then the two solids have equal volumes.
Think of a stack of coins: whether you stack them straight or lean them, the total volume is the same because each coin (cross-section) has the same area.
∎ This is why oblique prisms & cylinders have the same volume formula as right ones: V = Bh.
Worked Examples
Rectangular Prism
l = 8, w = 5, h = 4.
V = 160 cubic units
Cylinder
r = 4, h = 10.
V = 160π ≈ 502.7
Triangular Prism
Triangle base 6, triangle height 4, prism length 12.
B = ½(6)(4) = 12.
V = 144 cubic units
Common Pitfalls
Volume Units Are Cubed
Volume is in cubic units (cm³, in³), not squared. Three dimensions multiplied.
Real-Life Applications
Aquariums
Rectangular aquarium: V = lwh in cubic inches, then convert to gallons (1 gallon ≈ 231 in³).
Water Tanks
Cylindrical water tanks: V = πr²h determines capacity in gallons or liters.
Practice Quiz
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