---
book: 11
number: 33
id: "XI.33"
kind: "theorem"
uses: ["[[book-11/proposition-24]]", "[[book-6/proposition-1]]", "[[book-11/proposition-32]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.33"
license: "CC-BY-SA-4.0"
---

# XI.33

*Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.*

## Proof

Let *AB*, *CD* be similar parallelepipedal solids, and let *AE* be the side corresponding to *CF*; I say that the solid *AB* has to the solid *CD* the ratio triplicate of that which *AE* has to *CF*.

For let *EK*, *EL*, *EM* be produced in a straight line with *AE*, *GE*, *HE*, let *EK* be made equal to *CF*, *EL* equal to *FN*, and further *EM* equal to *FR*, and let the parallelogram *KL* and the solid *KP* be completed.

Now, since the two sides *KE*, *EL* are equal to the two sides *CF*, *FN*, while the angle *KEL* is also equal to the angle *CFN*, inasmuch as the angle *AEG* is also equal to the angle *CFN* because of the similarity of the solids *AB*, *CD*, therefore the parallelogram *KL* is equal ltand similargt to the parallelogram *CN*.

For the same reason the parallelogram *KM* is also equal and similar to *CR*, and further *EP* to *DF*; therefore three parallelograms of the solid *KP* are equal and similar to three parallelograms of the solid *CD*.

But the former three parallelograms are equal and similar to their opposites, and the latter three to their opposites; [[book-11/proposition-24|XI. 24]] therefore the whole solid *KP* is equal and similar to the whole solid *CD*. [[book-11/definitions#Definition 10|XI. Def. 10]]

Let the parallelogram *GK* be completed, and on the parallelograms *GK*, *KL* as bases, and with the same height as that of *AB*, let the solids *EO*, *LQ* be completed.

Then since; owing to the similarity of the solids *AB*, *CD*, as *AE* is to *CF*, so is *EG* to *FN*, and *EH* to *FR*, while *CF* is equal to *EK*, *FN* to *EL*, and *FR* to *EM*, therefore, as *AE* is to *EK*, so is *GE* to *EL*, and *HE* to *EM*.

But, as *AE* is to *EK*, so is *AG* to the parallelogram *GK*, as *GE* is to *EL*, so is *GK* to *KL*, and, as *HE* is to *EM*, so is *QE* to *KM*; [[book-6/proposition-1|VI. 1]] therefore also, as the parallelogram *AG* is to *GK*, so is *GK* to *KL*, and *QE* to *KM*.

But, as *AG* is to *GK*, so is the solid *AB* to the solid *EO*, as *GK* is to *KL*, so is the solid *OE* to the solid *QL*, and, as *QE* is to *KM*, so is the solid *QL* to the solid *KP*; [[book-11/proposition-32|XI. 32]] therefore also, as the solid *AB* is to *EO*, so is *EO* to *QL*, and *QL* to *KP*.

But, if four magnitudes be continuously proportional, the first has to the fourth the ratio triplicate of that which it has to the second; [[book-5/definitions#Definition 10|V. Def. 10]] therefore the solid *AB* has to *KP* the ratio triplicate of that which *AB* has to *EO*.

But, as *AB* is to *EO*, so is the parallelogram *AG* to *GK*, and the straight line *AE* to *EK* [[book-6/proposition-1|VI. 1]]; hence the solid *AB* has also to *KP* the ratio triplicate of that which *AE* has to *EK*.

But the solid *KP* is equal to the solid *CD*, and the straight line *EK* to *CF*; therefore the solid *AB* has also to the solid *CD* the ratio triplicate of that which the corresponding side of it, *AE*, has to the corresponding side *CF*.

Therefore etc. Q. E. D.

Porism. From this it is manifest that, if four straight lines be ltcontinuouslygt proportional, as the first is to the fourth, so will a parallelepipedal solid on the first be to the similar and similarly described parallelepipedal solid on the second, inasmuch as the first has to the fourth the ratio triplicate of that which it has to the second.