Determining Area and Volume



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: [pic]

• Mass: m

Helpful Equations for Circles and Cylinders:

[pic]

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.

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, [pic] and the height, h, of the cylinder.

[pic]

SolidWorks calculates Total Surface Area, S by adding all surfaces of a model. For a cylinder, there are three surfaces.

• Lateral surface area of the cylinder, SLAT

• Area of the top circle, [pic]

• Area of the bottom circle, [pic]

The area of the top circle equals the area of the bottom circle.

[pic]

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.

|Table 1 |

|Geometry: |Order: |Millimeter: (mm) |Inch: (in) |

|Radius |Linear |mm |in |

|Diameter |Linear |mm |in |

|Area |Squared |mm2 |in2 |

|Circumference |Linear |mm |in |

|Volume |Cubed |mm3 |in3 |

|Surface Area |Squared |mm2 |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.

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.

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?

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

• Area of the top ring, [pic]

• Area of the bottom ring, [pic]

The area of the top circle equals the area of the bottom circle.

[pic]

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?

|Example 1 Solution |Calculating Volume and Mass |

|Know: |Find: |

|[pic] |[pic] |

|Know: |Find: |

|[pic] |mass(g) |

|Model: |[pic] |

|Calculate: |[pic] |

|Example 1 Solution, cont |Calculating Surface Area of a Cylinder |

|Know: |Find: |

|[pic] |Total Surface Area, S(mm2) |

| | |

| | |

| | |

|Model: |[pic] |

|Calculate: |[pic] |

|Example 2 Solution |Calculating Volume and Mass of the Bracelet |

|Know: |Find: |

|[pic] |[pic] |

|Know: |Find: |

|[pic] |mass(g) |

|Model: |[pic] |

|Calculate |[pic] |

|Example 2 Solution |Calculating Surface Area of the Bracelet |

|continued | |

|Know: |Find: |

|[pic] |Total Surface Area, S(mm2) |

| | |

| | |

| | |

|Model: |[pic] |

|Calculate: |[pic] |

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.

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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Radius and Height of a Cylinder

[pic]

Top

SolidWorks calculates Total Surface Area

+

Outside + Inside +

Bottom

Mass Properties with Fillet Feature

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