EM - Cavalieri and Volume Lesson

Cavalieri and Volume

stacks of two coins, one stacked neatly and one with a lean Bonaventura Cavalieri was an Italian mathematician known for many great mathematical works. One of these is Cavalieri's Principle. In two dimensions, Cavalieri's Principle says that if two objects have the same length and the same width throughout, they have the same area. In three dimensions, it tells us that if two objects have the same height and the same cross-sectional area throughout, they have the same volume. If this sounds confusing, just imagine that you have two stacks of the same number of coins. One stack is perfectly vertical, while the other is slightly slanted. Regardless of whether or not the stacks are perfectly vertical, their volumes are the same because they still have the same number of coins. This is what Cavalieri's Principle tells us. We will explore this further!

two cylinders:
Cylinder 1: leaning with the equation A=20cm squared and a right angle labeled h=8cm
Cylinder 2=A=20cm squared
h=8cm

Notice the cylinders look very different in shape, but have the same base area and the same height. To calculate the volume of a cylinder, we multiply the area of the base times the height. Since both cylinders have a base area of 20cm² and a height of 8cm, they each have a volume of equation image indicator $LaTeX: 8cm\cdot20cm^2=160cm^3$ $8cm\cdot20cm^2=160cm^3$ .

A horizontal cross-section, or slice, of either of these cylinders would produce a circle; thus, we can think of a cylinder as a "stack" of an infinite number of these circular cross-sections. This is where the formula for the volume of the cylinder comes from! Check out the image and explanation to see how.

$LaTeX: V=A\left(x\right)\cdot h$ $V=A\left(x\right)\cdot h$

two clay cylinders with one showing the cross-section

But since equation image indicator $LaTeX: A\left(x\right)=\Pi r^2$ $A\left(x\right)=\Pi r^2$ , we know the volume of the cylinder is found by: $LaTeX: V=\Pi r^2h$ $V=\Pi r^2h$

This argument can be expanded to prisms of any base shape - rectangular prisms, triangular prisms, etc. All we must do is replace A(x) with the formula for the given base, and the horizontal cross-sections would be the shape of the given base instead.

two pyramids, with one leaning; both labeled ABCD with the height labelled as H

Since a square pyramid comprises $LaTeX: \frac{1}{3}$ $\frac{1}{3}$ equation image indicator of a square prism, it then follows that its volume should be 1/3 of the volume of the square prism. Thus, the volume can be found using the following formula: $LaTeX: V=\frac{1}{3}B\cdot h$ $V=\frac{1}{3}B\cdot h$ , where B = area of the base and h = height. In this case the base of each pyramid is a square whose area is s² , where s is the length of a side. Since each of these pyramids has a base area of s² and the same height, they have the same volume of $LaTeX: V=\frac{1}{3}s^2h$ $V=\frac{1}{3}s^2h$ equation image indicator by Cavalieri's principle.

one cone with a right angle drawn from the tallest point of cone to the bottom to a point on the base

By similar reasoning, we can find the formula for the volume of a cone. A cone is equation image indicator $LaTeX: \frac{1}{3}$ $\frac{1}{3}$ of a cylinder; thus, the volume of a cone is $LaTeX: V=\frac{1}{3}\Pi r^2h$ $V=\frac{1}{3}\Pi r^2h$ .

To learn what the formula for the volume of a sphere is and where it comes from, please watch the video below:

See the graphic below for a summary of the volume formulas you'll need to know! Make sure you copy them into your notes so you can refer to them as you practice and study.

image Volumes Formulas:
Prism. V=(Area of base)(height)
Cyldiner. V=Pi*r squared*h
Cone. V=1/3 * pi*r squared*height
Pyramid. V = 1/3(Area of base)(height)
Sphere V = 4/3 *pi*r to the third

Now, let's practice applying our volume formulas and Cavalieri's Principle!

IMAGES CREATED BY GAVS