NATIONAL
MATH + SCIENCE
INITIATIVE
Mathematics
Solids of
Revolution
I
Grade 8 as part of a unit on volumes of cylinders
and cones
MODULE/Connection to AP*
Areas and Volumes
*Advanced Placement and AP are registered trademarks
of the College Entrance Examination Board. The College
Board was not involved in the production of this product.
Modality
NMSI emphasizes using multiple representations
to connect various approaches to a situation in
order to increase student understanding. The
lesson provides multiple strategies and models
for using those representations indicated by the
darkened points of the star to introduce, explore,
and reinforce mathematical concepts and to enhance
conceptual understanding.
Objectives
Students will
● plot the points for a plane figure on a
coordinate plane.
● determine the area of that plane figure.
● draw and describe the solid formed by
revolving the plane figure about a vertical or
horizontal line.
● calculate the volume of the solid.
● compare the volumes determined when
revolving about different axes.
T e a c h e r
Level
n this lesson, students plot ordered pairs in the
first quadrant to create and determine the areas
of two-dimensional figures. The planar regions
are revolved about a horizontal or vertical line
to create three-dimensional figures, and students
calculate the volume for these solids. Students
transfer the dimensions of the planar region to the
solid as they identify the dimensions needed to
determine the volume. As students compare the
volumes when the same planar region is revolved
about different axes, they develop a deeper
conceptual understanding of how the change in the
radius and height impacts the overall volume of
a solid. The activity provides an engaging setting
for students to practice and apply their skills with
plotting points on a coordinate plane and calculating
areas and volumes.
P a g e s
About this Lesson
P
P – Physical
V – Verbal
G
V
A – Analytical
N – Numerical
G – Graphical
N
A
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i
Mathematics—Solids of Revolution
COMMON CORE STATE STANDARDS FOR
MATHEMATICAL CONTENT
This lesson addresses the following Common Core
State Standards for Mathematical Content. The
lesson requires that students recall and apply each
of these standards rather than providing the initial
introduction to the specific skill.
T e a c h e r
P a g e s
Targeted Standards
8.G.9: Know the formulas for the volume of cones,
cylinders, and spheres and use them to solve
real-world and mathematical problems.
See questions 1d, 1f, 2d, 2f, 3d, 3f, 4d, 4f, 5d,
6d, 6f, 6h
Reinforced/Applied Standards
6.G.3: Draw polygons in the coordinate plane given
coordinates for the vertices; use coordinates
to find the length of a side joining points
with the same first coordinate or the same
second coordinate. Apply these techniques
in the context of solving real-world and
mathematical problems.
See questions 1-6
6.G.1: Find area of right triangles, other triangles,
special quadrilaterals, and polygons by
composing into rectangles or decomposing
into triangles and other shapes; apply these
techniques in the context of solving realworld and mathematical problems.
See questions 1b, 2b, 3b, 4b, 5b, 6b
COMMON CORE STATE STANDARDS FOR
MATHEMATICAL PRACTICE
These standards describe a variety of instructional
practices based on processes and proficiencies
that are critical for mathematics instruction.
NMSI incorporates these important processes
and proficiencies to help students develop
knowledge and understanding and to assist them
in making important connections across grade
levels. This lesson allows teachers to address
the following Common Core State Standards for
Mathematical Practice.
MP.1: Make sense of problems and persevere in
solving them.
Students plot and connect points to create
2-dimensional figures on a coordinate plane,
determine the area of the planar region,
revolve the figure to form a 3-dimensional
solid, identify the solid, determine its
dimensions, and calculate the volume.
MP.3: Construct viable arguments and critique the
reasoning of others.
Students explain why the solids formed from
the same planar figure may have different
volumes.
MP.5: Use appropriate tools strategically.
Students create 3-dimensional models using
paper and pencil. Other visualization tools
can include physical or computer-generated
models.
MP.7: Look for and make use of structure.
Students associate the dimensions of the
planar figure to the radius and height of the
solid of revolution.
ii
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Mathematics—Solids of Revolution
The following skills lay the foundation for concepts
included in this lesson:
● Plot ordered pairs in the coordinate plane
● Calculate areas of rectangles and triangles
● Calculate volumes of cylinders and cones
MATERIALS AND RESOURCES
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●
●
●
●
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ASSESSMENTS
The following type of formative assessment is
embedded in this lesson:
● Students engage in independent practice.
T e a c h e r
The following additional assessments are located on
our website:
● Areas and Volumes – 7th Grade Free
Response Questions
● Areas and Volumes – 7th Grade Multiple
Choice Questions
●
Student Activity pages
Cut-out of planar region
Pencil, straw, or skewer
Tape
Scientific or graphing calculators
Graph paper
Mathematica demonstration simulating a
planar region revolved around either the
x or y axis to generate a 3-dimensional
figure located on the NMSI website with
resources for this lesson (a free download of
Mathematica Player is required).
P a g e s
Foundational Skills
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iii
Mathematics—Solids of Revolution
T e a c h e r
P a g e s
A
TEACHING SUGGESTIONS
lmost all students have difficulty in
visualizing 3-dimensional solids. Before
beginning the lesson, model the revolutions
around different axes by taping a cut-out of one
of the planar regions provided in the activity to a
pencil, straw, or skewer, and then rolling the pencil
between your hands to simulate the “sweeping out”
of the solid. Provide students with different shapes
and have them remove various tabs so that the solids
produced will have different radii and heights. Ask
students to describe the 3-dimensional shape that
is produced by revolving the figure along with the
radius and height of the solid. If students are having
difficulty identifying the solid when revolving the
shape, turn the manipulative upside down on the
table and twirl. An applet is listed in the resources
which demonstrates the solid formed when a planar
shape is revolved about either a horizontal or vertical
line.
The questions in this lesson build from cylinders and
cones to solids that are a combination of these two
solids. Carefully lead students through the steps for
drawing a solid of revolution.
●
●
●
●
●
●
iv
On the first grid, plot the ordered pairs.
Draw the boundary lines by joining the points
and determine the area.
On the second grid, redraw and shade the
planar region.
Reflect the shape about the indicated axis.
Connect the vertices and any other significant
points and their reflections with ellipses.
Refer students back to either their
manipulative or applet if they are having
trouble identifying the solid.
Questions 1 and 2 can be used as classroom
examples and require teacher guidance.
Question 3 can be used as a formative assessment.
These three questions ask students to compare the
volume of the solids that have been revolved about
both axes. Even though the planar regions are the
same, the solids have different radii and different
heights. Students are asked to explain why these
volumes are different. Using the formulas for the
two solids side-by-side may help them deepen their
understanding. Question 4 asks students to revolve
the planar region about a line instead of one of the
axes. Both questions 5 and 6 increase the level of
rigor by revolving two geometric figures about either
an axis or a line.
The instructions for calculating the volume of the
solids ask students to leave their answers in terms
of and also as a decimal answer rounded to three
decimal places. If a decimal approximation is used,
students should use the on their calculator instead
of using 3.14 for .
You may wish to support this activity with TINspire™ technology. See Calculations Using
Special Keys in the NMSI TI-Nspire Skill Builders.
Suggested modifications for additional scaffolding
include the following:
1 Provide the sketches of the regions and solids
for parts (a), (c), and (e).
Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Mathematics—Solids of Revolution
NMSI CONTENT PROGRESSION CHART
In the spirit of NMSI’s goal to connect mathematics across grade levels, a Content Progression Chart for
each module demonstrates how specific skills build and develop from sixth grade through pre-calculus in an
accelerated program that enables students to take college-level courses in high school, using a faster pace to
compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6
includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder
of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the
content from Grade 8 and all of the Algebra 1 content.
7th Grade
Skills/Objectives
Algebra 1
Skills/Objectives
Geometry
Skills/Objectives
Algebra 2
Skills/Objectives
Pre-Calculus
Skills/Objectives
Given 3 or
4 coordinate
points that form
a triangle or a
rectangle with
one side on a
horizontal line
and one side on
a vertical line,
calculate the area
of the figure.
Given 3 or
4 coordinate
points that form
a triangle or a
rectangle with
one side on a
horizontal line
and one side on
a vertical line,
calculate the area
of the figure.
Calculate the
area of a triangle,
rectangle,
trapezoid, or
composite of
these three figures
formed by linear
equations and/
or determine the
equations of the
lines that bound
the figure.
Calculate the
area of a triangle,
rectangle,
trapezoid, circle,
or composite
of these figures
formed by linear
equations or
equations of
circles and/or
determine the
equations of the
lines and circles
that bound the
figure.
Calculate the
area of a triangle,
rectangle,
trapezoid, circle,
or composite
of these figures
formed by linear
equations, linear
inequalities, or
conic equations
and/or determine
the equations
of the lines and
circles that bound
the figure.
Calculate the
area of a triangle,
rectangle,
trapezoid, circle,
or composite
of these figures
formed by linear
equations, linear
inequalities, or
conic equations
and/or determine
the equations
of the lines and
circles that bound
the figure.
Given 3 or
4 coordinate
points that form
a triangle or a
rectangle with
one side on a
horizontal or
vertical line,
calculate the
surface area
and/or volume
of the cone or
cylinder formed
by revolving the
bounded region
about either of the
lines.
Given 3 or
4 coordinate
points that form
a triangle or a
rectangle with
one side on a
horizontal or
vertical line,
calculate the
surface area
and/or volume
of the cone or
cylinder formed
by revolving the
bounded region
about either of the
lines.
Given the
equations of
lines (at least
one of which
is horizontal
or vertical.
that bound
a triangular,
rectangular,
or trapezoidal
region, calculate
the surface area
and/or volume of
the solid formed
by revolving
the region about
the line that is
horizontal or
vertical.
Given the
equations of
lines or circles
that bound
a triangular,
rectangular,
trapezoidal, or
circular region,
calculate the
volume and/or
surface area of
the solid formed
by revolving
the region about
a horizontal or
vertical line.
Given the
equations of lines
or circles or a set
of inequalities
that bound
a triangular,
rectangular,
trapezoidal, or
circular region,
calculate the
volume and/or
surface area of
the solid formed
by revolving
the region about
a horizontal or
vertical line.
Given the
equations of lines
or circles or a set
of inequalities
that bound
a triangular,
rectangular,
trapezoidal, or
circular region,
calculate the
volume and/or
surface area of
the solid formed
by revolving
the region about
a horizontal or
vertical line.
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T e a c h e r
6th Grade
Skills/Objectives
P a g e s
The complete Content Progression Chart for this module is provided on our website and at the beginning
of the training manual. This portion of the chart illustrates how the skills included in this particular lesson
develop as students advance through this accelerated course sequence.
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Mathematics—Solids of Revolution
T e a c h e r
P a g e s
vi
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NATIONAL
MATH + SCIENCE
INITIATIVE
Mathematics
Solids of Revolution
Answers
1. a.c. e.
d.
e.A cylinder with a radius of 3 and a height of 2
f.
larger than the solid formed
g.The solid formed by revolving about the y-axis is
by revolving about the x-axis. The radii and the heights of the cylinders are not the same when
revolved.
T e a c h e r
c.A cylinder with a radius of 2 and a height of 3
P a g e s
b.
2. a.c. e.
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vii
Mathematics—Solids of Revolution
b.
c.A cone with a radius of 3 and a height of 2
2
1
d.V = π (3 u ) (2 u ); V = 6 πu 3 ≈18.850 u 3
3
e.A cone with a radius of 2 and a height of 3
f.
larger than the volume when revolved
g.The volume when revolved about the x-axis is
about the y-axis. The radii and height of the cones are not the same when revolved.
T e a c h e r
P a g e s
3. a. c.e.
b. A =
1
5
1u )(5u )= u 2 = 2.5 u 3
(
2
2
c.A cone with a radius of 1 and height of 5
d.
e.A cone with a radius of 5 and a height of 1
f.
g.The volume of the cone is
larger when the region is revolved about the y-axis
than when it is revolved about the x-axis. The volumes are different because, when the regions are
revolved, the radii and the heights of the cones are not the same.
viii
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Mathematics—Solids of Revolution
4. a.
b.
d.
T e a c h e r
P a g e s
c.A cylinder with a radius of 3 and height of 5
e. A cylinder with a radius of 5 and a height of 3
f.
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ix
Mathematics—Solids of Revolution
5. a.
b.
T e a c h e r
P a g e s
c.A cone with a radius of 2 and a height of 2 on top of a cylinder with a radius of 2 and a height of 3
d.
6. a.
b.
x
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Mathematics—Solids of Revolution
c.A cone with a radius of 4 and a height of 2
d.
T e a c h e r
P a g e s
e.A cone with a radius of 2 and height of 4
f.
g.A cylinder with a radius of 2 and height of 4 with a cone of the same radius and height removed
from it.
h.
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xi
Mathematics—Solids of Revolution
7. a.Answers will vary. An example of a student answer could be (0, 0) (3, 0), and (0, 3).
T e a c h e r
P a g e s
b.Answers will vary. An example of a student answer could be (0, 0) (4, 0), and (0, 2).
c.A triangle will produce the same volume when revolved about the x- and y-axes if the height and
base of the triangle are on the axes and have the same dimension. A triangle will produce different
volumes when revolved about the x-and y-axes if the height and base of the triangle are on the axes
and have different dimensions.
xii
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NATIONAL
MATH + SCIENCE
INITIATIVE
Mathematics
Solids of Revolution
General instructions: When calculating volumes of cylinders and cones, give your answer both in terms
of and also as a decimal accurate to three decimal places. Use the key on your calculator and then
round your answer as the last step.
1. a. Draw line segments joining the points (0, 0), (0, 2), (3, 2), and (3, 0).
y
b. Calculate the area of the region formed.
x
c. Draw and describe the solid formed by revolving
the region about the x-axis.
y
x
d. Calculate the volume of the solid formed.
e. Draw and describe the solid formed by revolving
the region about the y-axis.
y
x
f. Calculate the volume of the resulting solid.
g. Compare the volumes in parts (d) and (f). Explain why these volumes are different.
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1
Mathematics—Solids of Revolution
2. a. Draw line segments joining the points (0, 0), (0, 3), and (2, 0).
b. Calculate the area of the region formed.
y
x
c. Draw and describe the solid formed by revolving
the region about the x-axis.
y
x
d. Calculate the volume of the solid formed.
e. Draw and describe the solid formed by revolving
the region about the y-axis.
y
f. Calculate the volume of the resulting solid.
g. Compare the volumes in parts (d) and (f). Explain why these volumes are different.
2
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x
Mathematics—Solids of Revolution
3. a. Draw line segments joining the points (0, 0), (0, 1), and (5, 0).
b. Calculate the area of the region formed.
y
x
c. Draw and describe the solid formed by revolving
the region about the x-axis.
y
x
d. Calculate the volume of the solid formed.
e. Draw and describe the solid formed by revolving
the region about the y-axis.
y
f. Calculate the volume of the resulting solid.
x
g. Compare the volumes in parts (d) and (f). Explain why these volumes are different.
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3
Mathematics—Solids of Revolution
4. a. Draw line segments joining the points (0, 0), (0, 3), (5, 3), and (5, 0).
b. Calculate the area of the region formed.
y
x
c. Draw and describe the solid formed by revolving
the region about the x-axis.
y
x
d. Calculate the volume of the solid formed.
e. Draw and describe the solid formed by
revolving the region about the vertical line x = 5.
y
f. Calculate the volume of the resulting solid.
4
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x
Mathematics—Solids of Revolution
5. a. Draw line segments joining the points (0, 0), (0, 5), (2, 3), and (2, 0).
b. Calculate the area of the region formed.
y
x
c. Draw and describe the solid formed by revolving
the region about the y-axis.
y
d. Calculate the volume of the solid formed.
Copyright © 2013 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
x
5
Mathematics—Solids of Revolution
6. a. Draw line segments joining the points (0, 0), (2, 4), and (2, 0).
b. Calculate the area of the region formed.
y
x
c. Draw and describe the solid formed by revolving
the region about the x-axis.
y
x
d. Calculate the volume of the solid formed.
e. Draw and describe the solid formed by revolving
the region about the vertical line x = 2.
y
x
f. Calculate the volume of the resulting solid.
g. Draw and describe the solid formed by revolving
the region about the y-axis.
y
h. Calculate the volume of the solid formed.
6
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x
Mathematics—Solids of Revolution
7.a. List three ordered pairs to create a triangular region that will produce the same volume when
revolved about the x- and y-axes.
y
x
b. List three ordered pairs to create a triangular region that will produce different volumes when
revolved about the x- and y-axes.
y
x
c. Compare these figures with a classmate. Write a general statement about the characteristics of the
triangles that produce the same volume when revolved about the x-and y-axes and the characteristics
of the triangles that produce different volumes when revolved about x- and y-axes.
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7
Mathematics—Solids of Revolution
8
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