---
book: 11
number: 29
id: "XI.29"
kind: "theorem"
uses: ["[[book-1/proposition-34]]", "[[book-1/proposition-8]]", "[[book-1/proposition-4]]", "[[book-1/proposition-36]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.29"
license: "CC-BY-SA-4.0"
---

# XI.29

*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 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 *AG*, *AF*, *LM*, *LN*, *CD*, *CE*, *BH*, *BK*, be on the same straight lines *FN*, *DK*; I say that the solid *CM* is equal to the solid *CN*.

For, since each of the figures *CH*, *CK* is a parallelogram, *CB* is equal to each of the straight lines *DH*, *EK*, [[book-1/proposition-34|I. 34]] hence *DH* is also equal to *EK*.

Let *EH* be subtracted from each; therefore the remainder *DE* is equal to the remainder *HK*.

Hence the triangle *DCE* is also equal to the triangle *HBK*, [[book-1/proposition-8|I. 8, 4]] and the parallelogram *DG* to the parallelogram *HN*. [[book-1/proposition-36|I. 36]]

For the same reason the triangle *AFG* is also equal to the triangle *MLN*.

But the parallelogram *CF* is equal to the parallelogram *BM*, and *CG* to *BN*, for they are opposite; therefore the prism contained by the two triangles *AFG*, *DCE* and the three parallelograms *AD*, *DG*, *CG* is equal to the prism contained by the two triangles *MLN*, *HBK* and the three parallelograms *BM*, *HN*, *BN*.

Let there be added to each the solid of which the parallelogram *AB* is the base and *GEHM* its opposite; therefore the whole parallelepipedal solid *CM* is equal to the whole parallelepipedal solid *CN*.

Therefore etc. Q. E. D.