Cup Roll Activity
Mathematical Goals: Teachers will be able to
 Accurately trace the path of a cup on paper and extrapolate on this process in
order to make connections to various geometric concepts and make
predictions.
Pedagogical Goals: Teachers will be able to
 See the applications that this or a similar activity would have in their
classroom. For example, this activity lends itself to a discussion of segments
of circles, tangent lines, triangle similarity, slant height and volume of cones
and truncated cones.
 Determine which of the Standards of Mathematical Practice are addressed in
this activity.
 Provide appropriate modifications to this activity for use in their classrooms.
Technological Goals:
 None
Alignment to the Common Core:
 G-GMD.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.
 G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel
line, and line segment, based on the undefined notions of point, line, distance
along a line, and distance around a circular arc.
 G-C.2 Identify and describe relationships among inscribed angles, radii, and
chords. Include the relationship between central, inscribed, and
circumscribed angles; inscribed angles on a diameter are right angles; the
radius of a circle is perpendicular to the tangent where the radius intersects
the circle.
 G-C.5 Derive using similarity the fact that the length of the arc intercepted by
an angle is proportional to the radius, and define the radian measure of the
angle as the constant of proportionality; derive the formula for the area of a
sector.
Mathematical Practices:
1 – Make sense of problems and persevere in solving them
2 – Reason abstractly and quantitatively
3 – Construct viable arguments and critique the reasoning of others
4 – Model with mathematics
5 – Use appropriate tools strategically
6 – Attend to precision
Length of Lesson: 90 minutes
Materials Needed:
Large paper
Millimeter rulers
Paper cups that are shaped like truncated cones
Compass
Pencil
Lipstick (to place on the cups so they can be traced more easily and
accurately)
Overview:
This exploratory activity has participants trace the path of a circular cup (which is
round on both the top and bottom, with the bottom circle having a smaller radius
than the top circle). Through this activity, participants can potentially discover the
connections to sectors of circles, similar triangles, tangent lines, slant height,
volumes of truncated cones and can make predictions based on their observations.
The problem:
Draw the circles created by tracing the path of a rolling cup on the paper. Determine
what mathematical concepts can be determined from this situation and explain the
relationships among the aspects identified.
Mathematical questions:
 How can we draw the entire circle by simply unfolding the paper cup and
tracing it only once – we are not allowed to lay the unfolded cup down
repeatedly. (We should draw lines congruent to two different tangents of the
arc that is drawn. The point of intersection of these two lines forms the
center of the larger circle.)
 Do the sectors formed by unfolding cups of different sizes always have the
same angle measurement? Why or why not? (No, they do not. The
difference in ratios of the bottom and top rims determine the angle formed.
 Using on the measurements from the cup, how can we determine the radius
of the large circle?
Pedagogical questions:
 What connections does this activity have to the theorems and concepts
dealing with circles, triangles and cones that are taught in the Common Core?
Extensions:
Provided a cup with different measurements, participants should make conjectures
about how the path of this cup will compare to the path of the first. Several cups
may be used and categorized.
Cut out a piece of paper (or provide a sketch containing the necessary
measurements) that will cover a lampshade exactly and to find the area of the piece
that you cut out. You are given that the lampshade has a 4.5 inch diameter at the
top, a 10.25 inch diameter at the bottom, and it is 6.5 inches tall.
An annulus has a 36 cm chord of the outer circle that is also tangent to the inner
concentric circle. Find the area of the annulus.
(
)
The area of an annulus is A = p R 2 - r 2 , where R is the radius of the large
circle and the radius of the small circle is r. Thus, using the Pythagorean
Theorem yields r 2 +182 = R2 .
r
R
18 cm
Thus, 182 = R2 - r 2 and, by substitution, A = 324p .
You have been tasked to determine the diameter of a broken shard of a cylindrical
bowl. You laid a meter stick across the shard and measured the arc to be 48 cm,
with a 6 cm depth from the midpoint of the chord. What was the bowl’s original
diameter?
Using the Pythagorean Theorem in a way that is similar to the previous
problem, we get
r 2 + 24 2 = ( r + 6 )
2
r 2 + 24 2 = r 2 +12r + 36
576 = 12r + 36
540 = 12r
45 = r
Thus, the diameter is 90 cm.