Cavalieri’s Principle

Table Of Contents

🎬 Math Angel Video: Cavalieri’s Principle Explained

An Example of Cavalieri’s Principle

Two equal-height stacks of 10 slices, one straight and one slanted, illustrating Cavalieri's Principle.

⏩️ (0:01)

🛎️ Imagine you have 10 slices of bread.

  • On the left, the slices are stacked neatly.
  • On the right, they are slanted.

Even though the shapes look different, the number and size of the slices are the same, so the total volume is unchanged.

This is a simple example to show how volume can stay the same, even if the shape looks different.

What is Cavalieri’s Principle?

Two identical stacks of bread slices illustrate Cavalieri's principle, showing that even if one stack is tilted, the total volume remains unchanged.

⏩️ (0:43)

🛎️ Definition of Cavalieri’s Principle:

Cavalieri’s principle states that if two solids have the same height and same cross-sectional areas, they have the same volume, even if they look different.

In other words, two solids have the same volume if:

  1. They have the same height, and
  2. At every level, their cross-sectional areas are the same.

 

🛎️ Applying Cavalieri’s Principle:

Cavalieri’s Principle can be very helpful. For example, both stacks have a 10 cm by 10 cm base and are 12 cm high.

The neat stack on the left has a volume of:

$$ \text{Volume} = (10 \text{ cm})^2 \times 12\text{ cm} = 1200 \text{ cm}^3$$

Here’s the clever part: by Cavalieri’s Principle, the slanted stack on the right must also have the same volume.

 

❇️ Exam Tip:

This principle gives us a shortcut, instead of working out the volume of a complicated or tilted solid directly, we can compare it to a simpler ones.

Using Cavalieri’s Principle in 3D

Cavalieri’s principle diagram comparing two solids with equal height and cross-sectional areas at every level, explaining they have the same volume.

⏩️ (1:33)

Cavalieri’s Principle doesn’t just work with stacks of slices, it also works for any 3D solids.

🛎️ What Cavalieri’s Principle Tells Us:

If two solids have the same height and the same cross-sectional area at every level, then their volumes are equal, even if the solids look very different.

🛎️ Using Cavalieri’s Principle for Solids:

In the picture, the two shapes both have the same height. At each level, their cross-sections match in area.

So, even though one looks curved and the other looks slanted, they must have the same volume.

🛎️ Why Cavalieri’s Principle Is Useful:

Cavalieri’s Principle is a powerful tool in geometry. It helps us compare and calculate volumes of unusual 3D shapes.

🍪 Quiz: Practice Cavalieri’s Principle in 3D Shapes

0%

Cavalieri's Principle

1 / 6

Q: If two solids have the same height and identical cross-sectional areas at the same height, what can we conclude about their volumes?

2 / 6

Q: A cylinder has a height of 9 cm and a cross-sectional area of $10 \, \text{cm}^2$. A rectangular prism has the same height of 9 cm and a cross-sectional area of $10 \, \text{cm}^2$. Do they have the same volume?

3 / 6

Q: Two solids have the same cross-sectional area of $20 \, \text{cm}^2$ at every height. The first solid has a height of 12 cm, and the second solid has a height of 15 cm. Do they have the same volume?

4 / 6

Q: Two solids have a height of 10 cm. One has a square cross-section with an area of $25 \, \text{cm}^2$, and the other has a circular cross-section with the same area of $25 \, \text{cm}^2$. What can you conclude about their volumes?

5 / 6

Q: Two solids have the same height of 9 cm. The first solid has a cross-sectional area of $10 \, \text{cm}^2$ at every height, and the second solid has a cross-sectional area of $15 \, \text{cm}^2$ at every height. Do they have the same volume?

6 / 6

Q: A stack of toast has a height of 15 cm, and its cross-sectional area is $40 \, \text{cm}^2$ at each level. Another stack has the same height but a cross-sectional area of $50 \, \text{cm}^2$ at each level. Using Cavalieri’s principle, what can we conclude about the volumes of these two stacks?

Your score is

The average score is 50%

0%

🎩 Stuck on Geometry Problems? Ask AI Math Solver

Need math help? Chat with our AI Math Solver at the bottom right — available 24/7 for instant answers.

5 1 vote
Article Rating
guest
0 Comments
Newest
Oldest Most Voted
Inline Feedbacks
View all comments

Leave a Comment

Your email address will not be published. Required fields are marked *