Unizor - Geometry3D - Pyramids - Volume of Any Pyramid

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Volume of Any Pyramids

In the previous lecture we have proved, based on the Cavalieri's principle, that the volume of a triangular pyramid equals to a product of the area of its base and one third of an altitude. Let's expand this to all other types of pyramids.

Let's consider a pyramid with any polygon as a base. For instance, hexagonal pyramid SABCDEF with apex S and hexagon ABCDEF as a base. To evaluate its volume, draw diagonals AC, AD and AE. Draw planes through each of these diagonals and apex S. These three planes dissect our pyramid into four triangular pyramids SABC, SACD, SADE and SAEF.

All these four triangular pyramids share the same altitude since their bases belong to the same plane and they have a common apex. Let the length of this altitude be h.
The volumes of these pyramids are:
VolumeSABC = AreaΔABC·h/3;
VolumeSACD = AreaΔACD·h/3;
VolumeSADE = AreaΔADE·h/3;
VolumeSAEF = AreaΔAEF·h/3;

Therefore, the volume of our original hexagonal pyramid equals to
VolumeSABCDEF =
(AreaΔABC+AreaΔACD+
+AreaΔADE+AreaΔAEF)·h/3 =
= AreaABCDEF·h/3

Again, we see that
THE VOLUME OF A PYRAMID EQUALS TO A PRODUCT OF THE AREA OF A BASE AND ONE THIRD OF THE ALTITUDE.

Example 1
Given a tetrahedron (right regular triangular pyramid with all edges of the same length) with the length of each edge d.
What is its volume?

Answer:
d³√2/12

Example 2

The base of the Egyptian (square) pyramid is a square with the length of a side d. The side edge of a pyramid is s.
Calculate the volume of this pyramid.

Answer:
[d²·√(s²−d²/2)]/3

Example 3

The base of the Egyptian (square) pyramid is a square with the length of a side d. The volume of a pyramid is V.
What is the length of its side edge?

Answer:
√[(3V/d)²+d²/2]

Example 4

A regular right hexagonal pyramid has an altitude equal to a radius of a circle that circumscribes a regular hexagon at its base and equals to d.
Calculate the volume of this pyramid.

Answer:
d³√3/2

Example 5

Given a triangular pyramid SABC (vertex S is an apex, triangle ΔABC is a base).
Let P be a midpoint of segment AB and Q be a midpoint of segment AC.
Draw a plane through points S, P and Q. It cuts our pyramid into two parts.
Find the ratio of their volumes.

Answer:
Ratio of a bigger part to a smaller is 3.

Example 6

Given a tetrahedron (right regular triangular pyramid with all edges of the same length).
What part of its volume is within an octahedron obtained by connecting all midpoints of its edges?


Answer:
One half.