Proving the Volume Formulas for Pyramids and Cones Using the Cavalieri Principle

The video tutorial explores the relationship between the volumes of pyramids and prisms, as well as cylinders and cones, using the Cavalieri principle. It demonstrates an informal proof showing that a pyramid's volume is one-third of a prism with the same base and height. By increasing the number of cross sections, the ratio of pyramid to prism volume approaches 1/3. The tutorial concludes that this principle also applies to cylinders and cones, reinforcing the reliability of the volume formulas.

What principle is used to compare the volumes of prisms and pyramids in this lesson?

• Pythagorean Theorem

• Cavalieri Principle

• Archimedes' Principle

• Pascal's Law

In the informal proof, what happens to the volume ratio as the number of layers increases?

• The ratio becomes unpredictable

• The ratio decreases

• The ratio remains constant

• The ratio increases

What is the approximate volume ratio of a pyramid to a prism when divided into five layers?

• 0.625

• 0.44

• 0.4688

• 0.5185

As the number of cross sections increases, what does the volume ratio approach?

• 0.25

• 0.5

• 0.75

• 0.3 repeating

What conclusion can be drawn about the difference between volume ratios as the number of layers increases?

• The difference decreases

• The difference becomes negative

• The difference remains constant

• The difference increases

How is the relationship between cylinders and cones similar to that of prisms and pyramids?

• Both have a 1/5 volume relationship

• Both have a 1/4 volume relationship

• Both have a 1/3 volume relationship

• Both have a 1/2 volume relationship

What geometric shape is a cone compared to in the lesson?

• A sphere

• A cube

• A cylinder

• A pyramid