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:
1
10
8
3
3
8
= 3.375.
= 10.125
• Radius of base: 10.125 = 2 𝜋 𝑟 ⇒ 𝑟 ≈ 1.61144.
My Measurements and Solution
3.375
1.61144
• Height of cone: 3.3752 − 1.611442 = ℎ2
• ℎ ≈ 2.96545
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:
𝜋 𝑟 2 = 𝜋 1.61144 2 ≈ 8.15790
• Side length of the square:
2.85620.
8.15790 ≈
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
ℎ,
5
2
ℎ,
5
1
ℎ,
2
3
ℎ,
5
4
ℎ
5
* 1/5 h is 1/5th of the height from the top for this
experiment.
Teamwork
To 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 Cone
1/5 h from the top marked
Slice of the Pyramid
1/5 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.
1
5
1.61144 = 0.32229
Cone
• 1/5 h from top
Radius of slice = 0.32229
Area of slice = π 0.32229
2
≈ 0.32632
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.
1
2.85620 = 0.57124
5
Pyramid
• 1/5 h from the top
Side length of slice = 0.57124
Area of slice = 0.57124 2 ≈ 0.32632
Position
Radius
Area of Slice of
Cone
Side
Area of Slice of
Pyramid
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
1
area is 𝐵ℎ ?
3
Calculus Connection: Cone Formula
The Formal Proof
Consider a cone with a base of
radius a and height h.
The cone will be created by rotating
𝒂
the line 𝒙 about the 𝑥-axis.
𝒉
Cone Formula
To find the exact volume of the cone,
start with the volume of one slice of
the cone perpendicular to the x-axis of
width Δ𝑥. For small Δ𝑥 this volume
will be approximately 𝜋
2
𝑎
𝑥 Δ𝑥,
ℎ
since the radius of an arbitrary slice is
𝑎
𝑥. Next, integrate this formula from
ℎ
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
𝑛
lim 𝜋
𝑛→∞
𝑖=1
ℎ
=𝜋
0
𝑎
𝑥𝑖
ℎ
Δ𝑥𝑖
𝑎2 2
𝑥 𝑑𝑥
2
ℎ
a2 1 3
= π 2
x
ℎ 3
=
2
𝜋 2
𝑎 ℎ
3
h
0
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
1
formula equal to 𝐵ℎ.
3
Formal Proof
Consider this structure, which is
a pyramid made up of three
steps. The top step has side
length 𝑎1 and the height is 𝑏1 .
The volume of this step is 𝑎12 𝑏1 .
http://mathforum.org/library/d
rmath/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
𝒂𝟐𝟏 𝒃𝟏 (𝟏𝟐 + 𝟐𝟐 + 𝟑𝟐 ).
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
𝑎22 𝑏2 .
The total volume of the figure is
𝒂𝟐𝟐 𝒃𝟐 (𝟏𝟐 + 𝟐𝟐 + ⋯ + 𝟔𝟐 ).
• If we continue this refinement, a pyramid with
𝑛 steps and a top block (the building block)
with volume 𝑎𝑛2 𝑏𝑛 , will have a total volume
of
𝒂𝟐𝒏 𝒃𝒏 (𝟏𝟐 + 𝟐𝟐 + ⋯ + 𝒏𝟐 ).
• Important Note: As 𝑛 increases, both the
length of the standard building block, 𝑎𝑛 , and
its height, 𝑏𝑛 , 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
𝟐 𝟐
𝒂𝒏 𝒏
= 𝑩 regardless of the number of
steps 𝑛.
(i.e. The building blocks have a base area of 𝑎𝑛2
and there are 𝑛2 building blocks on the bottom
level.)
Three Important Equalities
2. The height of the pyramid is 𝒃𝒏 𝒏 = 𝒉
regardless of the number of steps 𝑛.
Three Important Equalities
3. It is known that
𝟐
𝟐
𝟐
𝟏 +𝟐 + ⋯ + 𝒏 = 𝒏 𝒏 + 𝟏
𝟐𝒏+𝟏
𝟔
.
(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:
• 𝑎𝑛2 𝑏𝑛 12 + 22 + ⋯ + 𝑛2 =
• 𝑎𝑛2 𝑏𝑛 𝑛 𝑛 + 1
•
2𝑛+1
6
=
3
2
𝑛
𝑛
𝑛
2
• 𝑎𝑛 𝑏𝑛
+
+
=
3
2
6
1
1
1
2
3
• 𝑎𝑛 𝑏𝑛 𝑛
+
+ 2 =
3
2𝑛
6𝑛
1
1
1
2
2
𝑎𝑛 𝑛 𝑏𝑛 𝑛
+
+ 2
3
2𝑛
6𝑛
1
1
1
• 𝐵ℎ +
+ 2
3
2𝑛
6𝑛
=
Now we take the limit as 𝑛 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.
𝟏
𝟏
𝟏
𝟏
𝒍𝒊𝒎 𝑩𝒉 ( +
+
)
=
𝑩𝒉
𝒏→ ∞
𝟑 𝟐𝒏 𝟔𝒏𝟐
𝟑
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/62
3/why-is-the-volume-of-a-cone-one-third-ofthe-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_probl
em
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-theknot.org/Curriculum/Calculus/Cavalieri.shtml
Questions?