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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.

GeoB 11.8 Review Similar Cong

  • 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

GeoB 11.8 Similar Solids Ex

  • 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.

GeoB 11.8 NOT Similar Solids Ex

  • Cross-Multiplication

GeoB 11.8 Cross Mult

 

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.

GeoB 11.8 Scale Factor

 

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

GeoB 11.8 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

GeoB 11.8 Q1-01Answer: 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

GeoB 11.8 Q1-02

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.

GeoB 11.8 Q1-03

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.

GeoB 11.8 Q1-04

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.

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