HCPSS Worthwhile Math Task
How Much Ice Cream?
Common Core Standard
8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
The Task
You just placed a single scoop of delicious ice cream on a sugar cone. As you are about to take
the first bite, your mom asks if the ice cream can all fit inside the cone.
The ice cream cone has a radius of 3 cm and a height of 10 cm. If the scoop of ice cream is a
sphere with radius of 3.5 cm, will all of the ice cream fit inside the cone?
If so, how much more ice cream could the cone hold? If the cone is unable to hold all the ice
cream, design a new ice cream cone. Be sure to include how and why you chose the new
dimensions. Justify your answer with mathematics.
Facilitator Notes
1. Show a picture of an ice cream cone to class. (Clip Art)
2. This task is designed for groups of 3 to 4 to encourage discourse and encourage
collaboration.
3. Monitor students as they work in groups, posing questions to help them recall previous
material.
4. Answers will vary, they should agree that the ice cream will overflow and will need a
larger cone.
5. Encourage justification of the design of their sugar cone and share out to see the many
options that would work.
Follow-Up Questions
1. Which of the designs uses the smallest amount of material for the cone?
2. Can you design one that would use even less material?
3. Find the radius of the ice cream scoop that would fit the original cone with little to no
space remaining.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Solutions
The ice cream cone has a radius of 3 cm and a height of 10 cm. If the scoop of ice cream is a
sphere with radius of 3.5 cm, will the cone be able to hold all of the ice cream inside the cone?
Since the volume of the ice cream (sphere) is greater than the volume of the cone, the cone will
not be able to hold all the ice cream.
Volume of a cone
1
V  Bh
3
1
V  ( )(3) 2 (10)
3
V  94.2 cm 3

Volume of a sphere
4
V  r 3
3
4
V  ( )(3.5) 3
3
V  179.5 cm 3
 design a new ice cream cone. Be sure to include
If the cone is unable to hold all the ice cream,
how and why you chose the new dimensions.
Sample Responses:
1. Since the volume of a sphere is approximately 179.5 cubic centimeters, the new cone has to be
able to hold at least 180 cubic centimeters.
Keeping the radius of the cone the same, the new height of the cone has to at least 19.1 cm.
Therefore, the new cone should have the radius of 3 cm and height of 19.5 cm.
1
V  Bh
3
1
V  r 2 h
3
1
180  (3.14)(3) 2 h
3
180  9.42h
19.1 cm  h

We chose to make the ice cream cone with radius of 3 cm and height of 19.5 cm. We chose to
make the cone really long because sugar cones are absolutely delicious.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
2. If I change the radius of the cone to 3.25 cm, then the new height of the cone has to at least
16.3 cm. Therefore, I would make the new ice cream cone with a radius of 3.25 cm and height
of 16.5 cm.
1
V  Bh
3
1
V  r 2 h
3
1
180  (3.14)(3.25) 2 h
3
180  11.06h
16.3 cm  h

We decided to enlarge the radius of the cone by 0.25 cm and the height of the cone by 6.3 cm so
that the new ice cream cone is a little wider and longer.
3. I would redesign the cone by increasing the radius to 4 cm and height to 11 cm. Since the
radius of the sphere is 3.5 cm, I would make the radius of the cone to be a little wider.
1
V  Bh
3
1
V  r 2 h
3
1
180  (3.14)(4) 2 h
3
180  16.7h
10.8 cm  h

We chose to make the cone with radius of 4 cm and the height of 11 cm so that it can hold all the
ice cream. If the cone was made with 3 cm in radius, the height of the cone would have to be
over 19 cm and that’s not a reasonable size of the cone. We decided as a group that the radius of
4 cm and height of 11 cm for an ice cream cone was the most reasonable.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.