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1.2.1 How to use the principle The following example illustrates how to use the principle: Prove that any two pyramids with equal height and base area are of equal volume.
ܳ צ ܲ
൬ h ൬ h
൰ 2 ൰ 2
צ P צ P
ื P 1 ื P 2
ื ื S P 2
S P 1 ' S P 1
ۙۖۘ ۖۗ
' ~ P 1
Q Q
=
௩ ሳልልልልልሰܸ ଵ
֜ ܵ భ ′
ൌ ܵ మ ′
ൌ ܸ ଶ
H
S
P 2 '
' ~ P 2 S P 1 =
=
S 2
H
Let P and Q be parallel planes. The base of both pyramids are placed on Q. By
ܲ ଵ ′
צ ܲ ଵ and
ܲ ଶ ′
צ ܲ ଶ . Then we know that
ܲ ଵ ′ and
ܲ
ଵ are similar;
hypothesis,
as ܲ ଶ with the ratio of ܲ ଶ ′ and
ቀ h H
ቁ 2
. According to our assumption, S P 1 = S
. Therefore,
P 2
ൌ S
S
P 2 ' . These steps could be repeated for every plane parallel to Q. The
P 1 '
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