First, let's look at a more intuitive model, similar to what we did with pyramid building in class:
Next let's consider the basics of Cavalieri's principle:
Cavalieri's Principle states that if two or more objects have the same height and parallel cross sections at any level have the same area, then the objects have the same volume. One application of this is to extend the formula for the volume of a pyramid from triangular based pyramids (which can be derived directly by dissection), to pyramids with any shape base.
The volume of a triangular based pyramid is V = 1/3 A(base)*height. Since a pyramid of any base shape can be paired with a triangular base pyramid, and since it can be shown that the parallel cross sections at any height have equal area, the volume formula applies regardless of the shape of the base.
Cavalieri's Principle states that if two or more objects have the same height and parallel cross sections at any level have the same area, then the objects have the same volume. One application of this is to extend the formula for the volume of a pyramid from triangular based pyramids (which can be derived directly by dissection), to pyramids with any shape base.
The volume of a triangular based pyramid is V = 1/3 A(base)*height. Since a pyramid of any base shape can be paired with a triangular base pyramid, and since it can be shown that the parallel cross sections at any height have equal area, the volume formula applies regardless of the shape of the base.
Let's consider slant:
The volume of a prism as the area of the base time the height (V=Bh) can be demonstrated by moving the slider to the left and right. Cavalieri's principle (the volume of a prism stays the same no matter the slant as long as the height and area of the base remain the same) can be demonstrated by clicking and dragging the point labeled "shift".
The volume of a prism as the area of the base time the height (V=Bh) can be demonstrated by moving the slider to the left and right. Cavalieri's principle (the volume of a prism stays the same no matter the slant as long as the height and area of the base remain the same) can be demonstrated by clicking and dragging the point labeled "shift".
Next let's look at spheres: