Tuesday, September 15, 2015
Unizor - Geometry3D - Pyramids - Volume from Cavalieri Principle
Unizor - Creative Minds through Art of Mathematics - Math4Teens
Volume of Pyramids and Cavalieri's Principle
We will construct a prism from our pyramid that has the same base, altitude and an edge. Then we will show that our prism takes only one third of the volume of this prism.
Assume that SABC is a triangular pyramid with a base plane β that contains triangle ΔABC and apex (top vertex) S.
Let's add a few elements to this pyramid to make a prism.
Construct a plane γ parallel to plane β going through apex S of a pyramid. This plane will contain the top base of a prism we construct.
From points B and C we draw lines parallel to edge SA. These lines intersect with plane γ at points B' and C' correspondingly.
Consider an object bounded by two base planes β and γ and side faces SABB', SACC' and BB'C'C. Obviously, it's a triangular prism, we recommend to prove it as a self-study exercise.
All side faces of this prism are parallelograms.
Draw diagonal B'C in parallelogram BB'C'C.
Consider now two new triangular pyramids inside our prism:
CSB'C' with apex C and base SB'C' and
CSB'B with apex C and base SB'B.
These two pyramids, combined with our original pyramid SABC, dissect the prism into three pyramids. We will prove that the volumes of all three are the same and, therefore, the volume of our original pyramid equals to one third of the volume of prism with the same base and altitude.
First of all, consider pyramids SABC (original one with base ABC) and CSB'C'. They have the same altitudes (the distance between planes β and γ) and congruent bases ABC and SB'C'. Turning pyramid CSB'C' upside down, putting its base SB'C' on plane β and placing its apex into point S, we see two pyramids with congruent bases lying on the same plane and coinciding top vertices. This is exactly the situation discussed in Theorem B of the lecture Mini Theorems 1 of a previous topic 3-D Similarity. Therefore, the volumes of these two pyramids are equal.
Next, consider pyramids SABC (the original one) and CSBB'. Since any vertex in a triangular pyramid can be considered its apex, while the other three vertices form a base, consider our original pyramid SABC with an apex S and base ABC as a pyramid CSAB with apex C and base SAB. Now two pyramids CSAB and CSBB' have common apex C and their bases are two halves of a parallelogram ASB'B'. This is exactly the situation discussed in Theorem C of the lecture Mini Theorems 1 of a previous topic 3-D Similarity. Therefore, the volumes of these two pyramids are equal.
We see that, on one hand, the volume of the original pyramid SABC equals to the volume of pyramid CSB'C' and, on the other hand, the volume of that same pyramid (considered as CSAB with apex C) equals to the volume of pyramid CSBB'. Since these three pyramids combined form a prism, the volume of each pyramid is one third of the volume of a prism, which, as we know, equals to a product of the area of its base and the altitude.
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.