# Cavalieri's Principle Worksheets

What is Cavalieri's Principle? Geometry is an important branch of mathematics, and the concept of mensuration is one that we use in our daily lives.
When learning the concepts of solid geometry, there is one principle that all students need to learn. Originally known as the method of indivisibles, put forward by Bonaventura Cavalieri, an Italian mathematician.
The principle gives a separate statement for two-dimensional and three-dimensional shapes.
**Two-Dimensional Shapes** - Two-dimensional shapes are the ones that have a length and a width, but no depth. For such shapes, the principle states;
"Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas."
**Three-Dimensional Shapes** - Three-dimensional shapes are the ones that have a length, width, and depth. For such shapes, the principle states,
"Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes."
The principle puts light on the fact that different types of geometric shapes can have the same volume as well as area. On this page you will find a collection of worksheets and lessons that help you explore the uses of Cavalieri's Principle.

### Aligned Standard: High School Geometry - HSG-GMD.A.1

- Cross-section Step-by-step Lesson - Find the volume of the solid form.
- Guided Lesson - This equation always has been expressive for me. I have no idea how he figured this one out!
- Guided Lesson Explanation - The equation flows well as long as you know all of your variables.
- Practice Worksheet - I covered the two most common forms of question that you will see for this standard.
- Matching Worksheet - Find the volume and the cross-sectional size with this one.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

This is a very neat skill. It has many practical applications in biology.

- Homework 1 - This principle is often explored early in Calculus.
- Homework 2 - A huge benefit is that this formula can help you find the volume of an oblique prism.
- Homework 3 - The comparison presented here comes out by the technique of cross sectional stacking.

### Practice Worksheets

This skill is how they are starting to understand much more about the impact of concussions on athletes.

- Practice 1 - Calculate the volume of the with circular cross-sectional area.
- Practice 2 - Describe the area of a cross-section of a solid block at x(perpendicular to the x-axis). Find the volume of solid from x=0 to x=1.
- Practice 3 - Calculate the volume of a lunch box.

### Math Skill Quizzes

Make sure to work through the equation with students several times.

- Quiz 1 - This is sometimes called the mirror technique by some teachers.
- Quiz 2 - You can calculate some real unique measures here.
- Quiz 3 - This principle was formed by an Italian mathematician.

### Real World Examples of Cavalieri's Principle

For the most part, three-dimensional solid geometry that this principle is rooted in focuses on how this principle applies to standard geometric figures. Most of the real world is not formed into these perfect shapes so how are we to find value in studying this material unless we can apply it to our real lives? You surely can rest easy that this is truly worth your time because the applications to everyday life and comfort are bountiful. Most applications form around understand solid materials that we use on daily routine.

Civil engineers often use a series of calculations to determine the amount of materials that they will need to use based on the soil composition. These calculations are fundamental expansions on this principle. A more common example would be a deck of playing that are standing upright on table. You can also do this with a stack of coins. Have you ever seen the pipes in your bathroom or home? They are keen examples of Cavalieri's Principle because the volume of the pipe is same even went a pipe is bent into different directions and shapes. An even more modern adaptation of Cavalieri's Principle is its use with Magnetic Resonance Imaging (MRI) machines. MRI machines use magnetic fields and radio waves to create hyper-detailed images inside your body. Cavalieri's Principle is used to determine the exact sizes of tumors that pictured in the MRI images. This give physicians an ability to make a decision on treatment without having to cut the patient open for verification.