PROPOSITION 8.
Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.
Let there be similar and similarly situated pyramids of which the triangles
ABC,
DEF, are the bases and the points
G,
H the vertices; I say that the pyramid
ABCG has to the pyramid
DEFH the ratio triplicate of that which
BC has to
EF.
For let the parallelepipedal solids
BGML,
EHQP be completed.
Now, since the pyramid
ABCG is similar to the pyramid
DEFH, therefore the angle
ABC is equal to the angle
DEF, the angle
GBC to the angle
HEF, and the angle
ABG to the angle
DEH; and, as
AB is to
DE, so is
BC to
EF, and
BG to
EH.
And since, as
AB is to
DE, so is
BC to
EF, and the sides are proportional about equal angles, therefore the parallelogram
BM is similar to the parallelogram
EQ.
For the same reason
BN is also similar to
ER, and
BK to
EO; therefore the three parallelograms
MB,
BK,
BN are similar to the three
EQ,
EO,
ER.
But the three parallelograms
MB,
BK,
BN are equal and similar to their three opposites, and the three
EQ,
EO,
ER are equal and similar to their three opposites. [
XI. 24]
Therefore the solids
BGML,
EHQP are contained by similar planes equal in multitude.
Therefore the solid
BGML is similar to the solid
EHQP.
But similar parallelepipedal solids are in the triplicate ratio of their corresponding sides. [
XI. 33]
Therefore the solid
BGML has to the solid
EHQP the ratio triplicate of that which the corresponding side
BC has to the corresponding side
EF.
But, as the solid
BGML is to the solid
EHQP, so is the pyramid
ABCG to the pyramid
DEFH, inasmuch as the pyramid is a sixth part of the solid, because the prism which is half of the parallelepipedal solid [
XI. 28] is also triple of the pyramid. [
XII. 7]
Therefore the pyramid
ABCG also has to the pyramid
DEFH the ratio triplicate of that which
BC has to
EF. Q. E. D.
PORISM.
From this it is manifest that similar pyramids which have polygonal bases are also to one another in the triplicate ratio of their corresponding sides.
For, if they are divided into the pyramids contained in them which have triangular bases, by virtue of the fact that the similar polygons forming their bases are also divided into similar triangles equal in multitude and corresponding to the wholes [
VI. 20], then, as the one pyramid which has a triangular base in the one complete pyramid is to the one pyramid which has a triangular base in the other complete pyramid, so also will all the pyramids which have triangular bases contained in the one pyramid be to all the pyramids which have triangular bases contained in the other pyramid [
V. 12], that is, the pyramid itself which has a polygonal base to the pyramid which has a polygonal base.
But the pyramid which has a triangular base is to the pyramid which has a triangular base in the triplicate ratio of the corresponding sides; therefore also the pyramid which has a polygonal base has to the pyramid which has a similar base the ratio triplicate of that which the side has to the side.