---
book: 11
number: 30
id: "XI.30"
kind: "theorem"
uses: ["[[book-11/proposition-29]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.30"
license: "CC-BY-SA-4.0"
---

# XI.30

*Parallelepipedal solids which are on the same base and of the same height*, *and in which the extremities of the sides which stand up are not on the same straight lines*, *are equal to one another.*

## Proof

Let *CM*, *CN* be parallelepipedal solids on the same base *AB* and of the same height, and let the extremities of their sides which stand up, namely *AF*, *AG*, *LM*, *LN*, *CD*, *CE*, *BH*, *BK*, not be on the same straight lines; I say that the solid *CM* is equal to the solid *CN*.

For let *NK*, *DH* be produced and meet one another at *R*, and further let *FM*, *GE* be produced to *P*, *Q*; let *AO*, *LP*, *CQ*, *BR* be joined.

Then the solid *CM*, of which the parallelogram *ACBL* is the base, and *FDHM* its opposite, is equal to the solid *CP*, of which the parallelogram *ACBL* is the base, and *OQRP* its opposite; for they are on the same base *ACBL* and of the same height, and the extremities of their sides which stand up, namely *AF*, *AO*, *LM*, *LP*, *CD*, *CQ*, *BH*, *BR*, are on the same straight lines *FP*, *DR*. [[book-11/proposition-29|XI. 29]]

But the solid *CP*, of which the parallelogram *ACBL* is the base, and *OQRP* its opposite, is equal to the solid *CN*, of which the parallelogram *ACBL* is the base and *GEKN* its opposite; for they are again on the same base *ACBL* and of the same height, and the extremities of their sides which stand up, namely *AG*, *AO*, *CE*, *CQ*, *LN*, *LP*, *BK*, *BR*, are on the same straight lines *GQ*, *NR*.

Hence the solid *CM* is also equal to the solid *CN*.

Therefore etc. Q. E. D.