Volume of Spheres
— the volume of a sphere is four-thirds pi r cubed.
Introduction
Past Knowledge
Cylinder volume (12.3.1). Cone volume (12.3.2). SA of spheres (12.2.4).
Today's Goal
Apply V = ⁴⁄₃πr³ and find hemisphere volumes.
Future Success
Similar solids (12.3.4), calculus integration.
Key Concepts
Sphere Volume
Hemisphere Volume
Derivation
Cavalieri's Principle approach: A hemisphere has the same cross-sectional area at height h as a cylinder (radius r, height r) minus a cone inside it.
Full sphere = 2 hemispheres =
∎ Sphere = cylinder − cone (by Cavalieri) → ⁴⁄₃πr³.
Worked Examples
Sphere Volume
r = 6
V = 288π ≈ 904.8
Given Diameter
d = 14
r = 7.
V ≈ 1,436.8 cubic units
Find r from Volume
V = 36π. Find r.
→ →
r = 3
Common Pitfalls
r³ Not r²
Volume is (cubed), surface area is (squared). Cubing vs squaring makes a huge difference!
Real-Life Applications
Balloons
The volume of air in a spherical balloon: V = ⁴⁄₃πr³. The amount of helium needed scales with r³.
Earth's Volume
Earth ≈ sphere with r ≈ 6,371 km. V ≈ 1.08 × 10¹² km³.
Practice Quiz
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