PROOF 58

Prove.

The volume of a cone is 1/3 of the cylinder

In space.

What you may use

A1equal cross-sections ⟹ equal volumes
Cavalieri's principle (between the same two parallel planes)
A2volume of a prism = base area × height
volume of a prism
A3a non-overlapping division adds up
finite additivity of volume
A4moving a solid preserves volume
congruence invariance of volume
A5similarity ratio r ⟹ area ratio r²
Proof 54
A6a dilation by r centered at V exists
similarity dilation — Vf(P) = r·VP
A7the image under a dilation is similar with ratio r
property of dilations
A8a dilation sends planes to parallel planes
property of dilations
A9a plane cuts a convex polyhedron in two
cutting a polyhedron — both pieces are convex polyhedra
A10a diagonal bisects the area
parallelogram — the two triangles are congruent
A11the lateral faces of a prism are parallelograms
definition of a prism
A12a line meets a plane in one point
when the line does not lie in the plane
A13a triangle of area S exists
constructed with base b and height 2S/b
A14a(b+c) = ab + ac
distributive law
A15(ca)/c = a
cancellation (c ≠ 0)
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For a figure F in a plane α and a point V off α, the union of all segments joining V to the points of F is called the cone with base F and apex V.

Proof.