Having Fun with Geometry

Cavalieri’s Principle

MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Cavalieri’s Principle

Historical Note

Bonventura Francesco Cavalieri (1598 – 1647)

Cavalieri’s work with indivisibles was a precursor to the development of calculus. His method of indivisibles is what is now known as Cavalieri’s Principle.

Activity: Developing a Formula for the Volume of a Cone

• Items needed– A partner (let’s work in pairs!)– Ruler– Tape– Scissors– Bag of dry beans– Cone stencil– 2 x sheets of card stock– 12” string

Step One: The Cone• Cut out the cone stencil and create a cone by

taping the solid edge to the dashed edge.• Use the string to determine the circumference of

the base of the cone in inches.• Measure the slant height of the face of the cone

in inches.• Use these two measurements to determine the

height of the cone in inches. (5 decimal places)

My Measurements and Solution

• Round to five decimal places for accuracy.

• Slant height of the face of the cone:

• Circumference of base:

• Radius of base:

My Measurements and Solution

• Height of cone:

3.375

1.61144

Step Two: The Cylinder

Construct a cylinder that has the same height and circumference as the cone.

*** Remember to include an extra half of an inch for overlap before cutting out your cylinder stencil. ***

Step Two: The Cylinder

Tape the cylinder to a piece of cardstock so that it won’t slide when being filled.

Step Three: The Beans

• Fill the cone completely and accurately with beans.

• Pour them into the cylinder.• Repeat until the cylinder is full.

What do you observe?

Creating the Formula

• What is the formula for the volume of a cylinder?

• What is the formula, based on your experiment, for the volume of a cone?

Activity: Comparing the Volume of a Cone and a Pyramid with the Same Base Area and Height

Activity: Comparing the Volume of a Cone and a Pyramid with the Same Base Area and Height

Step One

Using the radius of the base of the cone in the previous activity, construct a square that has the same area as the base of the cone.

Solution Based on My Measurements

• Area of the base of the cone:

• Side length of the square:

Step Two

• Visualize the pyramid that can be constructed with this square as its base, having the same height as the cone.

Optional activity: Have students create the pyramid and confirm that the cone and the pyramid can hold the same volume of beans.

Question

We know the base of the cone and the base of the pyramid have the same area, but if we slice the cone and pyramid at a different height in a plane parallel to the base, will these slices have the same area?

Let’s Experiment!

Comparing slices cut by a parallel plane

Comparing Slices

Compare a slice of the cone to a slice of the pyramid at the following heights in planes parallel to the base.

* 1/5 h is 1/5th of the height from the top for this experiment.

TeamworkTo save time, each table will compute the area of a slice of the cone and a slice of the pyramid at one height.

Table 1 = 1/5 h (from the top)

Table 2 = 2/5 h

Table 3 = 1 /2 h

Table 4 = 3/5 h

Table 5 = 4/5 h (close to bottom)

Results

Place your results on the board

Solutions Based on My Measurements

Slice of the Cone1/5 h from the top marked

Slice of the Pyramid1/5 h from the top marked

Cone

• 1/5 h from top

To find the length of the

base of the smaller triangle,

multiply the radius of the cone

by 1/5.

= 0.32229

Cone

• 1/5 h from top

Radius of slice = 0.32229

Area of slice =

Pyramid

• 1/5 h from the top

To find the length of the

base of the smaller triangle,

multiply the side length of

the base of the pyramid by 1/5.

Pyramid

• 1/5 h from the top

Side length of slice

Area of slice =

Position Radius Area of Slice of Cone

Side Area of Slice ofPyramid

1/5 0.32229 0.32632 0.57124 0.32632

2/5 0.64458 1.30528 1.14248 1.30526

1/2 0.80572 2.03947 1.42810 2.03947

3/5 0.96686 2.93682 1.71372 2.93684

4/5 1.28915 5.22104 2.28496 5.22104

base 1.61144 8.15790 2.85620 8.15788

Cavalieri’s Principle

• In the three-dimensional case: If two regions are trapped between two parallel planes (imagine one at the top of the cone and pyramid, and one at the bottom), and every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volume.

• Reference: See Wikipedia – You will find some cool mathematics there.

Cavalieri’s Principle

• Activity: Take your pennies and stack them neatly on your desk.

• Think about the volume of this stack of pennies.• Change the stack so that it is disturbed. (i.e.

Push it in the middle, but don’t knock it down.)• Does this collection of pennies still have the

same volume?• Yes! This is Cavalieri’s Principle in action.

Cavalieri’s Principle

Cavalieri’s Principle

• How does Cavalieri’s Principle apply to our cone and pyramid?

• What can we say about the formula for the volume of a pyramid given that the formula for the volume of a cone of equal height and base area is

Calculus Connection: Cone FormulaThe Formal Proof

Consider a cone with a base of radius a and height h.

The cone will be created by rotating the line about the -axis.

Cone FormulaTo find the exact volume of the cone, start with the volume of one slice of the cone perpendicular to the x-axis of width. For small this volume will be approximately, since the radius of an arbitrary slice is . Next, integrate this formula from 0 to h (using the Fundamental Theorem of Calculus to move from a sum of volumes of slices to the exact volume), to capture the volume of the entire cone. 

Cone Formula

                                     

Pyramid Formula

The formal proof of the formula for the volume of a pyramid is given in the notes for this lesson.

The short story … Start with a step-pyramid and find its volume by computing the volume of each step.

Pyramid Formula

Then you refine the pyramid, giving it more steps.

Each time the base area and height are the same.

Pyramid Formula

In limit, as the number of steps goes to infinity, we have a pyramid with a volume formula equal to

Formal Proof

Consider this structure, which is a pyramid made up of three steps. The top step has side length  and the height is .  

The volume of this step is . 

http://mathforum.org/library/drmath/view/53646.html 

Formal Proof

The next step has four blocks the same size as the top step and the bottom step has nine.  The formula for the volume of the entire shape is  .

Now consider a refinement of the previous pyramid. This step-pyramid has six steps, with the same base area as the previous pyramid, and the same height. The volume of the top step is .

The total volume of the figure is .

• If we continue this refinement, a pyramid with steps and a top block (the building block) with volume , will have a total volume of

.

• Important Note: As increases, both the length of the standard building block, and its height, decrease. However, the base area and the height are fixed throughout the refinement process.

Three Important Equalities

1.  The area of the base of the pyramid is regardless of the number of steps .

(i.e. The building blocks have a base area of and there are building blocks on the bottom level.)

 

Three Important Equalities

2. The height of the pyramid is regardless of the number of steps .

Three Important Equalities

3. It is known that

.

(Recall the story of Gauss adding the numbers from 1 to 100 when he was in elementary school by developing a formula.)

Our total volume formula may be transformed as follows:

Now we take the limit as goes to infinity, which mathematically represents the process of continuing to refine our step pyramid until it morphs into a smooth pyramid with a square base and triangular sides.

Rather Use Applets Than Beans?

• Scrolling down this page you will see an animated demonstration of the cube with three pyramids being placed consecutively inside it. This is followed by a discussion with pictures of the cone and pyramid volumes being compared and Cavalieri’s Principle being used.

• http://math.stackexchange.com/questions/623/why-is-the-volume-of-a-cone-one-third-of-the-volume-of-a-cylinder

Rather Use Applets Than Beans?

• This site shows the three pyramids that will fit into a cube and how these pyramids can be transformed into a right square pyramid with the same volume using Cavalieri’s Principle. This is followed by a comparison of a cone to the pyramid and the resulting formula.

• http://nrich.maths.org/1408

Supplemental Material

The Napkin Ring Problem

http://en.wikipedia.org/wiki/Napkin_ring_problem

Supplemental Material

The relationship between the formula for a cylinder, cone and sphere:

• Given a cylinder and cone with the same radius and height, r and h respectively, such that r is equal to h, and a sphere of radius r = h, the volume of the cylinder minus the volume of the cone is equal to the volume of ½ of the sphere (i.e. the hemisphere).

• http://www.cut-the-knot.org/Curriculum/Calculus/Cavalieri.shtml

Questions?