Similar Solids with a uniform dilation (scale factor)


Similar Solids Introduction
Goal: Identify the relationship between1, 2 and 3 dimensions with a give scale factor.

Task 1 - Make a prism and use the scale tool to perform a uniform dilation.
Screencast - How to use the Scale Tool


external image 8sBysYf_5ycFI_ufbTagMEF8WHzpuam6Hj4BFQlNgmXLo7RTC19WT3ltbFB51vFj-DcA4t0pwyaIHLHDRd4VnFH65NiGVAqAoFDpzfaSO5wC5d0qi4gThis cube shows the Scale Tool reference points. The cube can be scaled in many different directions depending on which reference points you select. For example, you can stretch or shrink the heigh only. This does not result in similar figures. Why not?
When working with this tool, be sure that you use one of the options that begins with the word Uniform as the points are selected and shown in red.

Getting Started
A 2m x 3m x 1.5m prism (shown below) was created with one vertex at the origin. The figure on the right is a copy of the prism that was dilated with a scale factor of 0.5. The side dimensions of the copy are 1 x 1.5 x .75 meters. Compare the area of the faces shown in red. Is the area half as big? Compare the volumes of the two prisms. Do you see a relationship? Is the volume half as big?

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Task 2 - Create similar prisms.
Create a prism, make three copies and use the Scale Tool to perform dilations on each of the three copies. You may use tools in SketchUp to gather information about the linear measurements, surface area, and volume of the new solids. You may use a calculator as necessary. Making sketches on paper is recommended.

Step 1. Open the SketchUp file for Figure 2 (Creating Solids Assignment) OR create a prism that is 3x4x5 feet.

Step 2. Create a copy of your shape and paste it next to the original. Dilate with a scale factor of 2. I suggest you color one side of each shape blue. For example, you can color the top face that is 3 x 4 blue. Be consistent!

Step 3. Repeat step 2, but use a scale factor of 3. Repeat one more time with a scale factor of 4.
Similar Prisms.png
Step 4. Compare the dimensions, surface area and volume between the original prism to dilated prisms. Use the table below to guide your work.
Uniform Dilation
Original
Prism 2
Prism 3
Prism 4
Scale Factor
1
2Double each linear dimension
3Triple each linear dimension
4 Quadruple each linear dimension
Length of base




Width of base




Height




Area




Volume






Step 5. Look at the solids and estimate the ratio between the sides (length of base to length of base). Do the same for area and then for volume between each pair of prisms. Use the table below to guide your work:
Screen Shot 2014-11-27 at 9.49.37 AM.png
Step 5. Make a conjecture regarding the relationship between one dimension (side length), two dimension (area) and three dimensions (volume).


Task 3 - Test your conjecture with a cylinder (not a polyhedron).
The graphic below shows a cylinder with a radius of 2.5 meters and height of 4 meters. Next to the original shape are two copies. The linear measurements were first doubled and then tripled.
similar cylinders.png

Step 1. Create you own cylinders in a SketchUp file. I used darker colors as the scale factor increased.
Step 2. Compare the linear dimensions, surface area and volume between the original cylinder to new cylinders. Use the table below to guide your work.
Uniform Dilation
Original
Cylinder 2
Cylinder 3
Cylinder 4
Scale Factor
1
2
Double each linear dimension
3
Triple each linear dimension
4
Quadruple each linear dimension
Radius of Base




Height




Area of Base




Volume




Step 3. Look at the solids and estimate the ratio between the radii of the bases. Do the same for area and then for volume. Use the table below to guide your work:
Screen Shot 2014-11-27 at 9.39.56 AM.png
Step 4. Make a conjecture regarding the relationship between one dimension (ratio), two dimension (area of base) and three dimensions (volume) for the cylinder.




Task 4 - Test your conjecture with a pyramid.
Screen Shot 2014-11-27 at 9.33.39 AM.png


Task 5 - Test your conjecture with a sphere.
Screen Shot 2014-11-27 at 9.36.45 AM.png
Task 2-5 Worksheet

Prism and Cylinder Measurements

Updated December 2014