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Cavalieri's Principle

HSG-GMD.A.1
Answer Keys Here

Aligned To Common Core Standard:

Geometric Measurement - HSG-GMD.A.1

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.

Printable Worksheets And Lessons




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.