Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights

Theorem

In the words of Euclid:

Cones and cylinders which are on equal bases are to one another as their heights.

(The Elements: Book $\text{XII}$: Proposition $14$)

Proof

Let $EB$ and $FD$ be cylinders on equal bases, the circles $AB$ and $CD$.

It is to be shown that:

$EB : FD = GH : KL$

where $GH$ and $KL$ are the axes of $GH$ and $KL$.

Let the axis $KL$ be produced to the point $N$.

Let $LN = GH$.

Let the cylinder $CM$ be described about the axis $LN$.

We have that:

$EB$ and $CM$ have the same height

and

$EB$ and $CM$ are on equal bases.

From Proposition $11$ of Book $\text{XII} $: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases:

$EB$ and $CM$ are equal.

We have that the cylinder $FM$ has been cut by the plane $CD$ which is parallel to its bases.

Therefore by Proposition $13$ of Book $\text{XII} $: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis:

$CM : FD = LN : KL$

But:

$CM = EB$

and:

$LN = GH$

Therefore:

$EB : FD = GH : KL$

But from Proposition $10$ of Book $\text{XII} $: Volume of Cone is Third of Cylinder on Same Base and of Same Height:

$EB : FD = ABG : CDK$

where $ABG$ and $CDK$ are cones whose bases are $AB$ and $CD$ and whose apices are $G$ and $K$.

Therefore:

$GH : KL = ABG : CDK$

and:

$GH : KL = EB : FD$

$\blacksquare$

Historical Note

This proof is Proposition $14$ of Book $\text{XII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XII}$. Propositions