Assignment
• P. 822-825: 1, 2, 321 odd, 24, 26, 32,
33, 35, 36
• P. 832-836: 1, 2-24
even, 28, 30-36,
40, 41
• Challenge
Problems
Units, Units2, and Units3
Recall that length is
measured in units:
1
1
Length: 1 unit
1
And area is measured
in square units:
Area: 1 square unit
Units, Units2, and Units3
The volume of
something (a
polyhedron, a room,
a bottle) is
measured in cubic
units: in3, ft3, cm3,
m3, etc. It’s a threedimensional
measurement.
1
1
1
Volume: 1 cubic unit
Volume
Volume is the measure
of the amount of
space contained in a
solid, measured in
cubic units.
– This is simply the
number of unit cubes
that can be arranged
to completely fill the
space within a figure.
Exercise 1
Find the volume of
the given figure in
cubic units.
12.4-12.5: Volume of Prisms, Cylinders,
Pyramids, and Cones
Objectives:
1. To derive and use the formulas for the
volume of prisms, cylinders, pyramids,
and cones
Volume Postulates
Volume of a Cube
– The volume of a cube is V = s3.
Volume Congruence
– If two polyhedra are congruent, then their
volumes are equal.
Volume Addition
– The volume of a solid is the sum of the
volumes of all of its nonoverlapping parts.
Investigation 1
In this Investigation,
you will begin by
examining the
volumes of simple
rectangular solids.
You will then
generalize your
observations to
apply to other kinds
of solids.
Investigation 1
Step 1: Find the volume of each right
rectangular prism. (How many cubes
measuring 1 cm on an edge will fit into
each solid?)
Investigation 1
Step 2: To get the volume of the prism, you
could use a principle of multiplication to
find the number of cubes:
Area of the base, B
Number of cubes in the base = (2)(4) = 8 cubes
Height of prism, h
Since the prism is 3 layers high, V = (8)(3) = 24 cubes
Exercise 2
Use the formula for the volume of a prism to
help derive a formula for the volume of a
cylinder with radius r and a height h.
Volume of Prisms and Cylinders
Volume of a Right
Prism
Volume of a Right
Cylinder
V r h
V  Bh
•
•
B = area of the base
h = height of prism
2
•
•
r = radius of cylinder
h = height of cylinder
Exercise 3
Find the volume of the
regular hexagonal
prism shown.
Exercise 4
The rectangle shown can be rotated around the yaxis or the x-axis to make two different solids of
revolution. Which solid would have the greater
volume?
Exercise 5
Find the volume of the solid of revolution formed by
revolving the rectangle shown around the y-axis.
Sections
When a solid is cut by a plane, the resulting
plane figure is called a section. A section
that is parallel to the base is a crosssection.
Exercise 6
Exercise 6
Cavalieri’s Principle
Suppose you wanted to find the volume of an
oblique rectangular prism with a base 8.5
inches by 11 inches and a height of 6
inches…
Cavalieri’s Principle
The shape of the oblique rectangular prism
can be approximated by a slanted stack of
three reams of 8.5” x 11” paper…
Cavalieri’s Principle
The shape can be even better approximated
by the individual pieces of paper in a
slanted stack…
Cavalieri’s Principle
Rearranging the paper formed into an
oblique rectangular solid back into a right
rectangular prism changes the shape, but
does it change the volume?
Cavalieri’s Principle
Similarly, you could
use a stack of coins
to show that an
oblique cylinder has
the same volume as
a right cylinder with
the same base and
height.
Cavalieri’s Principle
If two solids have the same height and the
same cross-sectional area at every level,
then they have the same volume.
All 3 of these shapes have the same volume.
Exercise 7
Name the solid
shown, and then
find its volume.
Exercise 8
Given the dimensions shown
in the diagram, how much
concrete would be used to
make 20 cinderblocks?
Exercise 9
The volume of the cylinder is
3148 yd3. Find the length
of the radius.
Investigation 2
In this Investigation you will
discover the relationship
between the volumes of
prisms and pyramids with
congruent bases and the
same height and between
cylinders and cones with
congruent bases and the
same height.
Investigation 2
Step 1: Choose a prism and a
pyramid that have congruent
bases and the same height.
Step 2: Fill the pyramid, then
pour the contents into the
prism. About what fraction of
the prism is filled by the volume
of the volume of one pyramid?
Step 3: Check your answer by repeating
Step 2 until the prism is filled.
Investigation 2
Step 4: Choose a cone and a
cylinder that have congruent
bases and the same height
and repeat Steps 2 and 3.
Step 5: Did you get similar
results with both your
pyramid-prism pair and the
cone-cylinder pair?
Volume of Pyramids and Cones
Volume of a Pyramid
Volume of a Cone
1
V  Bh
3
•
•
B = area of the base
h = height of pyramid
1 2
V  r h
3
•
•
r = radius of cone
h = height of cone
Exercise 10
Find the volume of the solid of revolution
formed by rotating the triangle around the
y-axis.
Exercise 11
Find the volume of the solid of revolution
formed by rotating the triangle around the
y-axis.
Exercise 12
You are using the funnel shown to
measure the coarseness of a
substance. It takes 2.8 seconds for the
substance to empty out of the funnel.
Find the flow rate of the substance in
mL per second (1 mL = 1 cm3).
Exercise 13
Find the volume of the composite figure.
Assignment
• P. 822-825: 1, 2, 321 odd, 24, 26, 32,
33, 35, 36
• P. 832-836: 1, 2-24
even, 28, 30-36,
40, 41
• Challenge
Problems