# 11.8 – Similar Solids

## Key Terms

• Proportion – An equation stating that two ratios are equal.
• A proportion of the form $\frac{a}{b}=\frac{c}{d}$ is true if the cross products are equal: ad=bc.
• Similar – Having exactly the same shape.
• When figures are similar, corresponding angles are congruent and corresponding sides are proportional in length.
• The symbol ~ means “is similar to.”

## Review

Remember
•  Similar Polygons
• have the exact same shape.
• have corresponding angles that are congruent.
• have corresponding sides proportional in length.
• Line Segments
• are all similar because they all have the same shape.
• they are not all congruent because they can have different lengths
• All congruent figures are similar, but not all similar figures are congruent.

• Translations
• When you translate a figure, you move it along a straight line (without changing its shape).
• Dilation
• When you dilate a figure, you make it larger or smaller (without changing its shape).

## Notes

Similarity, Ratios, and Cross-Multiplication
• The ratios of two solids’ dimensions MUST be similar for the two solids to be similar.
• Dimensions are linear measurements: edge lengths, widths, heights, slant heights, radii, diameters, perimeters, or circumferences.
• Similar
• Both have a ratio of 3 to 1

• NOT Similar
• They have different ratios.  The left has a ratio of 3 to 1 and the right has a ratio of 4 to 1.

• Cross-Multiplication

Scale Factor
• The ratio between corresponding lengths of solid figures is constant.
• Surface Area: ratios are SQUARED edges
• If an edge is 3 feet, the squared edge would be $3^{2}$ feet; so, 9 feet.
• Volume: ratios are CUBED edges
• If an edge is 3 feet, the cubed edge would be $3^{3}$ feet; so, 27 feet.

Scale Modeling
• The scale model has a scale factor of $\frac{1}{25}$ and is similar to the actual building.
• If the height of the actual building will be 20 feet, what is the height of the scale drawing?
• Setup: $\frac{1}{25}=\frac{x}{20}$
• Cross-Multiply: $20=25x$
• Answer: $0.8=x$
•  Scale Model

## Examples

 Ex 1. The two cones below are similar. What is the height of the larger cone? Setup: Cross-multiply $\frac{x}{7}=\frac{5}{4}$ $35=4x$ Answer: $x=\frac{35}{4}$ Ex 2. The two prisms below are similar. What is the value of x? Setup: Cross-multiply $\frac{2}{10}=\frac{1}{x}$ $2x=10$ Answer: $x=5$ Ex 3. The two solids are similar, and the ratio between the lengths of their edges is 2:7. What is the ratio of their surface areas? Setup: Area is squared, so square the ratio. Answer: $2^{2}\colon 7^{2}=4\colon 49$ Ex 4. If the ratio between the radii of the two spheres is 3:7, what is the ratio of their volumes? Setup: Volume is cubed, so cube each number in the ratio. Answer: $3^{3}\colon 7^{3}=27\colon 343$ Ex 5. If two pyramids are similar and the ratio between the lengths of their edges is 3:11, what is the ratio of their volumes? Setup: Volume is cubed, so cube each number in the ratio. Answer: $3^{3}\colon 11^{3}=27\colon 1331$ Ex 6. It is true that All spheres are similar. Ex 7. It is true that the ratio of the volumes of two similar solid polyhedra is equal to the cube of the ratio between their edges. Ex 8. It is false that the ratio of the volumes of two similar solid polyhedra is equal to the square of the ratio between their edges. Remember, solids volumes are CUBED! Ex 9. It is true that the ratio of surface areas of two similar solids is equal to the square of the ratio between their corresponding edge lengths.