PROOF 158

Prove.

The volume of a sphere

In space.

What you may use

A1equal cross-sections ⟹ equal volumes
Cavalieri's principle — between the same two parallel planes
A2volume of a cylinder = base area × height
volume of a cylinder
A3volume of a cone = ⅓ × base area × height
Proof 58
A4cross-section area of a cone = ρ²S
lemma of Proof 58 — ρ is the ratio of distances measured from the apex
A5area of a circle = πr²
Proof 50
A6in a right triangle a²+b²=c²
Pythagorean theorem (Proof 56)
A7non-overlapping pieces add
additivity of volume and area
A8moving a solid preserves volume
congruence invariance of volume
A9ℓ⊥α ⟹ ℓ⊥ lines in α
perpendicularity of a line and a plane
A10x ≥ 0 ⟹ (√x)² = x
definition of the square root
A11x,y≥0: x ≤ y ⟺ x² ≤ y²
monotonicity of squaring (Proof 104)
A12cylinder cross-section ≅ base
definition of a cylinder — cross-sections parallel to the base
A13a(b − c) = ab − ac
distributive law
A14associativity, commutativity, cancellation
arithmetic of the reals
A15x ≤ y ⟹ x + c ≤ y + c
property of inequalities — transposition
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

Take the plane α to be z = 0 and a point O on it as the origin. The part of the ball with center O and radius r where z ≥ 0 is called the hemisphere H.

Proof.