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

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## Theorem

In the words of Euclid:

(*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.

- $EB$ and $CM$ are equal.

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

- $CM : FD = LN : KL$

But:

- $CM = EB$

and:

- $LN = GH$

Therefore:

- $EB : FD = GH : KL$

- $EB : FD = ABG : CDK$

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