Algebra and Geometry Applications for STEM
Determine the Radius, Volume and Density of a Cylinder:
Calculating the volume in a design environment allows the designer to determine the
overall dimensions of the product and the amount of material required to manufacture the
product.
Required Variables:
 Radius: r
 Diameter: d
 Area: A
 Circumference: C
 Height: h
 Volume: V
 Total Surface Area: S
 Lateral Surface Area: SLAT
 Density: 
 Mass: m
Radius and Height
of a Cylinder
Helpful Equations for Circles and Cylinders:
d
r
2
A  r 2
C  2r
V  r 2 h
S LAT  2rh

m
V
Dimension a Circle in SolidWorks:
In SolidWorks, circles use diameter dimensions to determine size. Arcs use radius
dimensions to determine size. Diameter dimensions are preceded with the diameter
symbol, Ø. The radius dimensions are preceded with the R symbol.
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Algebra and Geometry Applications for STEM
Mathematical formulas use the variable r. Use the variable, r in
this text to calculate the radius of a circle.
Dimension a Cylinder in SolidWorks:
An Extruded Feature adds height to a circle. Height is displayed as
a linear dimension. The linear dimension is represented by h. Use
the variable, h in this text to calculate the height of a cylinder.
Calculating Surface Area of a Cylinder in SolidWorks:
Lateral Surface Area is the area
around a cylinder. Lateral
Surface area is defined the
product of the circumference of
the circle, 2r and the height, h,
of the cylinder.
S LAT  2rh
SolidWorks calculates Total
Surface Area, S by adding all
surfaces of a model. For a
cylinder, there are three surfaces.



2r
Lateral surface area of the cylinder, SLAT
Area of the top circle, A  r 2
Area of the bottom circle, A  r 2
The area of the top circle equals the area of the bottom circle.
S  S LAT  2 A
S  2rh  2(r 2 )
S  2rh  2r 2
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Algebra and Geometry Applications for STEM
Units of Measurement:
Determine units at the start of a design problem.
State the units throughout the problem.
Common units of length are millimeters and inches.
Geometry:
Radius
Diameter
Area
Circumference
Volume
Surface Area
Order:
Linear
Linear
Squared
Linear
Cubed
Squared
Table 1
Millimeter: (mm)
mm
mm
mm2
mm
mm3
mm2
Inch: (in)
in
in
in2
in
in3
in2
SolidWorks Units:
 Set the units for the document using the
Tools, Options, Document Properties, Units
from the Main menu.

Determine units at the start of a design
problem. Measure the radius and diameter in
linear units such as inches, feet, millimeters or
meters.
Example 1: Determine the Volume, Mass and
Surface Area of a Cylinder:
Part 1: Create the Cylinder.
1.) h of the Cylinder: 10mm.
2.) d of the Cylinder: 75mm.





Start a SolidWorks Session.
Sketch a circle on the Front Plane.
Dimension the circle 75mm.
Create an Extruded Boss/Base Feature.
Extrude the Sketch. Use 10mm for Depth.
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Algebra and Geometry Applications for STEM
Part 2: Determine the Mass Properties
of a Cylinder:
 Apply a Material. Apply Rubber.
 Click Tools, Mass Properties from the
Main menu.
 View the Density, Mass, Volume and
Surface Area of the Cylinder.
Part 3: Determine the analytical
solution without SolidWorks: Use the
supplied equations.
1.) Determine the following:
o Radius of the circle.
o Height of the cylinder.
o Volume of the cylinder.
o Mass of the cylinder.
o Surface Area of the cylinder.
Example 2 Determine the Volume, Mass and
Surface Area of the Bracelet.
The copper Bracelet is designed with an inside
diameter of 65mm and an outside diameter of
75mm.
Part 1: Create the Bracelet.
1.) ID of the Cylinder: 65mm.
2.) OD of the Cylinder: 75mm.









Start a SolidWorks Session.
Sketch a circle on the Front Plane.
Dimension the circle 75mm.
Create an Extruded Boss/Base Feature.
Extrude the Sketch. Use 10mm for Depth.
Sketch a circle on the Front face.
Dimension the circle 65mm.
Create an Extruded Cut Feature.
Use the Through All option.
Part 2: Determine the Mass
Properties of a copper Bracelet:
 Apply a Material. Apply Copper.
 Click Tools, Mass Properties from
the Main menu.
 View the Density, Mass, Volume
and Surface Area of the Cylinder.
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Algebra and Geometry Applications for STEM
Part 3: Determine the analytical solution without SolidWorks: Use the supplied
equations.
2.) Determine the following:
o Radius of the inside circle, r1.
o Radius of the inside circle, r2.
o Height of the cylinder.
o Volume of the cylinder.
o Mass of the cylinder.
o Surface Area of the cylinder.
If you the Fillet feature to your bracelet, the
analytical mass properties differ from the
SolidWorks calculated mass properties.
Why?
Mass Properties with Fillet Feature
Determine Surface Area of the Bracelet:
SolidWorks calculates Total Surface Area, S by adding all surfaces of a model. For the
Bracelet, there are four surfaces.



Lateral surface area of the outside cylinder, S1
Lateral surface area of the outside cylinder, S2
2
2
Area of the top ring, A   ( r2  r1 )

Area of the bottom ring, A   ( r2  r1 )
2
2
The area of the top circle equals the area of the bottom circle.
S  S1 S 2  2 A
S  2r1 h  2r2 h  2 ( r2  r1 )
2
2
S  2h ( r1  r2 )  2 ( r2  r1 )
2
2
Top
Outside
+
+
Inside
+
Bottom
SolidWorks calculates Total Surface Area
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Algebra and Geometry Applications for STEM
Part 4: Determining Cost of the Mold Base.
The retail company, XYZ-Jewelry, Inc. requires a steel mold base to be designed and
manufactured to create the bracelets. XYZ-Jewelry receives the following quotes from
their suppliers:
Design Time:
20 hours
Machine Shop Time: 40 hours
The rate for Design Time is $200/hour. The rate for Machine Shop Time is $150/hour.
The Machine Shop Time includes the material required to create the mold.
The XYZ-Jewelry estimated $10,000 for the mold base. Is this a good cost estimate?



What is the cost for the Design Time?
What is the cost for the Machine Shop Time?
What is the total cost for XYZ-Jewelry to obtain the mold base?
Part 5: Determining Break Even Cost of the Bracelet.
An agent at XYZ-Jewelry needs to know how many bracelets the company must sell in
order to make a profit. The agent estimates the mold base to be $10,000. Each bracelet
costs $.10 to manufacture, package and ship. The bracelet sells for $1.00 each.


Determine the profit for each bracelet?
Determine the number of bracelets for the company to make a profit after
paying for the mold base?
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Algebra and Geometry Applications for STEM
Example 1 Solution
Calculating Volume and Mass
Know:
Find:
radius ( mm)
d  75 mm
h  10mm
Volume( mm 3 )
mass( g )
  .001g / mm3 for rubber
Surface Area ( mm 2 )
Model :
V  (r 2 h )mm 3
Calculate :
r  75 / 2  37.5mm
V  (r 2 h )mm 3
V   (37.5mm) 2 (10mm)
V   (37.52 mm 2 )(10mm)
V   (1406.25)mm 2 (10)mm
V  3.14(14062.5)mm 3
V  44156.3mm 3
Know:
V  44156.3mm3
Find:
mass(g)
  .001g / mm3 for rubber
Model:
 
m
V
 (V ) 
V  m
m  V
Calculate:
m
(V )
V
m  V
g
( 44156.3mm 3 )
3
mm
m  44.16 g
m  .001
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Algebra and Geometry Applications for STEM
Example 1 Solution, cont
Calculating Surface Area of a Cylinder
Know:
r  75 / 2  37.5mm
Find:
Total Surface Area, S(mm2)
h  10mm
Model:
S  S LAT  2 A
S  2rh  2(r 2 )
S  2rh  2r 2
Calculate:
S  2rh  2r 2
S  2 ( rh  r 2 )
S  2 (( 37.5mm)(10mm)  (37.5mm) 2 )
S  2 (375mm 2  1406.3mm 2 )
S  2 (1781.3mm 2 )
S  11186.3mm 2
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Algebra and Geometry Applications for STEM
Example 2 Solution
Calculating Volume and Mass of the Bracelet
Know:
ID  65mm
Find:
inside radius , r1
OD  75 mm
outside radius , r2
Volume( mm 3 )
h  10mm
mass( g )
  .0089 g / mm for copper
Surface Area ( mm 2 )
Model :
V  (r2 h  r1 h )mm 3
3
2
2
V  h ( r2  r1 )mm 3
2
Calculate :
2
r1  65 / 2  32.5mm
r2  75 / 2  37.5mm
V  h ( r2  r1 )mm 3
2
2
V   (10)( 37.52  32.52 )mm 3
V  10 (1406.25  1056.25)mm 3
V  (10)( 3.14)( 350)mm 3
V  10990mm 3  11000mm 3
Know:
V  11000mm3
Find:
mass(g)
  .0089 g / mm3 for rubber
Model:
m
V
m  V
Calculate
m  V

m  .0089
g
(11000mm 3 )
3
mm
m  97.9 g
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Algebra and Geometry Applications for STEM
Example 2 Solution
continued
Know:
r1  65 / 2  32.5mm
Calculating Surface Area of the Bracelet
Find:
Total Surface Area, S(mm2)
r2  75 / 2  37.5mm
h  10mm
Model:
S  S1 S 2  2 A
S  2r1h  2r2 h  2 ( r2  r1 )
2
2
S  2h( r1  r2 )  2 ( r2  r1 )
2
Calculate:
2
S  2h( r1  r2 )  2 ( r2  r1 )
2

2
S  2 h( r1  r2 )  ( r2  r1 )
2
2


S  2 (10mm)( 70mm)  (1406.3  1056.3)mm )
S  2 (700mm )  (350mm )
S  2 1050mm 
S  2 (10mm)( 37.5mm  32.5mm)  (( 37.5mm) 2  (32.5mm) 2 )

2
2
2
2
S  2(3.14)(1050)mm 2
S  6594mm 2
Review Problem Solving Steps:
 Break up larger problems into smaller ones.
 State your known variables and what you have to find.
 State units of measurement for each variable.
 Create an algebraic model. State units in the model.
 Factor variables to simplify the equation.
 Substitute values.
 Calculate in steps. Use parenthesis.
 State units for the answer.
 Check answer. Compare results in SolidWorks.
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Algebra and Geometry Applications for STEM
Solution Part 4: Determining Cost of the Mold Base.
The retail company, XYZ-Jewelry, Inc. requires a steel mold base to be designed and
manufactured to create the bracelets. XYZ-Jewelry receives the following quotes from
their suppliers:
Design Time:
20 hours
Machine Shop Time: 40 hours
The rate for Design Time is $200/hour. The rate for Machine Shop Time is $150/hour.
The Machine Shop Time includes the material required to create the mold.
The XYZ-Jewelry estimated $10,000 for the mold base. Is this a good cost estimate?
a) What is the cost for the Design Time?
20 hours*($200/hour) = $4000
b) What is the cost for the Machine Shop Time?
40 hours*($150/hour)= $6000
c) What is the total cost for XYZ-Jewelry to obtain the mold base?
Total Cost = $4000+$6000 = 10000
Solution Part 5: Determining Break Even Cost of the Bracelet.
An agent at XYZ-Jewelry needs to know how many bracelets the company must sell in
order to make a profit. The agent estimates the mold base to be $10,000. Each bracelet
costs $.10 to manufacture, package and ship. The bracelet sells for $1.00 each.


Determine the profit for each bracelet?
Determine the number of bracelets for the company to make a profit after paying
for the mold base?
Profit per bracelet: $1.00 - $.10 = $.90.
Total Cost: $10000
Let x = number of bracelets
Total Cost = Profit * x
10000 = .90x
x = 11111.1, on the 11112 bracelet the company will make a profit.
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