InterMath | Workshop Support | Write Up Template
Title
Changing Cubes
Problem Statement
What happens to the surface area of a cube when all the dimensions are doubled? tripled?
What happens to the volume of a cube when all the dimensions are doubled? tripled?
If you want to increase the volume of a cube by 216 cubic units, how should you change the
dimensions of the cube?
Repeat the investigation above using a rectangular prism and a cylinder.
Problem setup
Determine what happens to the surface area and volume of a cube when the side is doubled or
tripled. Then do the same thing for a rectangular prism and cylinder.
Plans to Solve/Investigate the Problem
First construct the net for a cube. Then calculate the area and volume of the cube. Then double
the edge of the cube and recalculate the area and volume. Then triple the edge of the cube and
again recalculate the area and volume. Then construct the net for a rectangular prism and follow
the same procedure. Finally construct the net for a cylinder and again follow the same
procedure.
Investigation/Exploration of the Problem
I first constructed a net for a cube of edge 1. I measured all edges and calculated the total
surface area and volume.
BC = 1.00 cm
CD = 1.00 cm
DE = 1.00 cm
EF = 1.00 cm
FG = 1.00 cm
GH = 1.00 cm
HI = 1.00 cm
IJ = 1.00 cm
NO = 1.00 cm
OB = 1.00 cm
NC = 1.00 cm
MD = 1.00 cm
S urface Area
V olume
6.00 c m2
1.00 c m3
B
LE = 1.00 cm
LI = 1.00 cm
EH = 1.00 cm
JK = 1.00 cm
LM = 1.00 cm
KL = 1.00 cm
MN = 1.00 cm
C
O
N
F
D
M
E
G
H
L I
K
J
I then constructed a net for a cube of edge 2. Again I measured all edges and calculated the total
surface area and volume.
DE = 2.00 cm
EF = 2.00 cm
F
FG = 2.00 cm
GH = 2.00 cm
G
B
V olume
24 cm2
8 cm3
C
HI = 2.00 cm
D
IJ = 2.00 cm
E
JK = 2.00 cm
H
KL = 2.00 cm
LM = 2.00 cm
S urface Are a
O
N
MN = 2.00 cm
M
NO = 2.00 cm
L
I
OB = 2.00 cm
NC = 2.00 cm
MD = 2.00 cm
K
LE = 2.00 cm
J
LI = 2.00 cm
EH = 2.00 cm
Then I constructed a net for a cube of edge 3.
B
BC = 3.00 cm
CD = 3.00 cm
DE = 3.00 cm
EF = 3.00 cm
S urface Are a
V olume
54 c m2
27 c m3
C
FG = 3.00 cm
GH = 3.00 cm
O
F
HI = 3.00 cm
D
IJ = 3.00 cm
JK = 3.00 cm
KL = 3.00 cm
N
G
LM = 3.00 cm
E
MN = 3.00 cm
NO = 3.00 cm
M
OB = 3.00 cm
NC = 3.00 cm
H
MD = 3.00 cm
L
LE = 3.00 cm
LI = 3.00 cm
EH = 3.00 cm
Surface Area = 54 cm2
Volume =
27 cm3
I
K
J
I noticed that when an edge was doubled, the total surface area was multiplied by four and the
volume was multiplied by 8. Also, when an edge was tripled, the total surface area was
multiplied by 9 and the volume was multiplied by 27.
I then decided to use a spreadsheet to look for patterns of what happened as the cube is doubled
or tripled. Then I looked at what happened as the rectangular prism is doubled or tripled.
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I noticed that when the edge was doubled, the surface area multiplied by 4, and the volume
multiplied by 8. I also noticed when the edge was tripled; the surface area multiplied by 9 and
the volume was multiplied by 27. I concluded when the edge was increased by n, the resulting
surface area was multiplied by n2 and the resulting volume was multiplied by n3.
I then decided to explore what happened to a cylinder when the dimensions were doubled or
tripled.
Radius
Height
Total Surface
Area
Volume
1
2
3
4
5
3
6
9
12
15
21.98
87.92
197.82
351.68
549.5
9.42
75.36
254.34
602.88
1177.5
I noticed that as with the rectangular prism and cube, when the dimensions of the cylinder were
doubled, the surface area was multiplied by four and the volume was multiplied by eight. I
noticed that when the dimensions were tripled, the surface area was multiplied by nine and the
volume multiplied by 27. I noticed that when the dimensions of the cylinder were multiplied by
n, the surface area was multiplied by n2 and the volume by n3.
Extensions of the Problem
Find how much you should change the dimensions of a cube, if you want the volume to increase
by 216 cubic units. I realized that since I did not know the original dimensions of the edge, I
needed to find what number I needed to multiply the dimension by for the volume to increase by
216 cubic units. If the original dimension was x, then the volume would be x3. I wanted to find
the dimension for the figure such that the volume would now be x3 + 216. That would mean that
if I multiplied the original dimension by n, the resulting volume would be (x * n)3 and this would
equal x + 216. If you solve for n, then n 
3
3
x 3  216
. I proved my results using the
x
spreadsheet below.
Edge of Cube
Volume
Factor to
increase the
edge by
3
1
2
4
5
6
1.5
2.5
3.5
4.5
5.5
27
1
8
64
125
216
3.375
15.625
42.875
91.125
166.375
2.080083823
6.009245007
3.036588972
1.635533155
1.397273606
1.25992105
4.020725759
2.456528472
1.820938701
1.499313815
1.319675459
New
Volume
Change in
Volume
243
217
224
280
341
432
219.375
231.625
258.875
307.125
382.375
216
216
216
216
216
216
216
216
216
216
216
Author & Contact
Debbie Brigham
Link(s) to resources, references, lesson plans, and/or other materials
Changing Cube
Changing Cube Spreadsheet