# 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.

This 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 height 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?

Task 2 - Create similar prisms.
Create a prism, make three copies and use the Scale Tool to perform a dilatation 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 Activity) 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.

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 eachlinear dimension 4Quadruple eachlinear dimension Length of base Width of base Height Area Volume

Step 5. Look at the solids and estimate the ratio between the sides (e.g. length of base of Prism 1 to length of base Prism 2). Do the same for area and then for volume between each pair of prisms. Use the table below to guide your work:

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.

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 2Double eachlinear dimension 3Triple eachlinear dimension 4Quadruple eachlinear 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:

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.

Task 5 - Test your conjecture with a sphere.

Task 2-5 Worksheet

Prism and Cylinder Measurements

Updated December 2014