As with planar figures, 3-dimensional figures can be similar as long as the definition of similarity is met. Consider the two right square prisms below.

To determine whether or not the solids are similar, verify both conditions of similarity.

Condition 1: All corresponding angles are congruent.

• Since the base is a square, what type of angles are all of the base angles?
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The interior angles of a square are all right angles, so the base angles of the square prisms are all right angles.
• Since the prisms are right prisms, what is the angle between the bases and the height of the prisms?
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The angle between the bases and the height of a right prism is a right angle.
• Are all corresponding angles congruent?
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Yes. All right angles are congruent, so the corresponding angles are also all congruent.

Condition 2: All corresponding side lengths are proportional.

• A prism has three dimensions: length, width, and height. Match the ratios of the corresponding dimensions, then check to see if the ratios are equal to each other.
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small height large height = 3 in. 6 in. = 1 2
small length large length = 2 in. 4 in. = 1 2
small width large width = 2 in. 4 in. = 1 2
Since the ratios are all equal to 1 2 , they are equal to each other.

• What is the scale factor that could be used to generate the larger prism from the smaller prism?
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The scale factor is the number that should be multiplied by each dimension of the smaller prism to generate the dimensions of the larger prism. In this case, that scale factor is 2.

The two cylinders shown are similar. Use the measurements shown to determine the height of the smaller cylinder.

Step 1: Identify the scale factor used to generate the smaller cylinder from the larger cylinder. Use a pair of corresponding measures that you know to set up a ratio. Drag the measurements from the figure to the appropriate place in the ratio below.

Step 2: Use the ratio from step 1 to set up a proportion containing the missing dimension that you are trying to find. Drag the appropriate measures to the proportion boxes shown below.

Step 3: Now that you have two equal ratios, you have a proportion. Solve the proportion.

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The height of the smaller cylinder is 5 yards.