Congruent and Similar Solids

Similar Solids

• Definition: Solids are similar if their corresponding measurements are proportional.
  • Example: For cones, ( \frac{h_1}{h_2} = \frac{r_1}{r_2} ) where ( h ) is height and ( r ) is radius.
• Comparison to Similar Shapes: Similar solids behave like similar triangles/polygons.

Congruent Solids

• Definition: Solids are congruent if all corresponding measurements are equal.
  • Example: ( \frac{H_1}{H_2} = \frac{L_1}{L_2} = 1 ) for heights ( H ) and slant heights ( L ).

Ratios in Similar Solids

• Ratio of Side Lengths (A to B): ( \frac{A}{B} )
• Ratio of Surface Area: ( \left(\frac{A}{B}\right)^2 )
• Ratio of Volume: ( \left(\frac{A}{B}\right)^3 )

Practical Examples

1. Determine Similarity or Congruency:

   • Compare corresponding sides and check if ratios are equal.
   • Example: Solids with side ratios ( \frac{2}{6} ), ( \frac{4}{12} ), ( \frac{3}{9} ) simplifies to ( \frac{1}{3} ) indicating similarity.
   • Scale Factor for similar solids: 1:3.

2. Determine Congruency:

   • Example with radius: Both radii 5m, thus ( \frac{5}{5} = 1 ), indicating congruency.

3. Volume Ratios for Similar Pyramids:

   • Given heights: 4 in and 7 in.
   • Scale factor: ( \frac{4}{7} ).
   • Ratio of volumes: ( \left(\frac{4}{7}\right)^3 = \frac{64}{343} ).

4. Find Volume of Similar Cylinders:

   • Smaller cylinder: Radius 2.5 in, Height 10 in.
   • Larger cylinder: Height 16 in, find radius and volume.
   • Calculate scale factor: ( \frac{10}{16} = \frac{5}{8} ).
   • Ratio of volumes: ( \left(\frac{5}{8}\right)^3 = \frac{125}{512} ).
   • Volume of smaller cylinder: ( 62.5\pi ).
   • Calculate volume of larger cylinder using proportions, result: 84.2 in³.

Conclusion

• Understanding ratios in similar solids allows for comparison of attributes like surface area and volume.
• Utilize scale factors and proportional reasoning to solve problems involving similar and congruent solids.

Homework

• Begin working on section 12.8 homework exercises related to congruent and similar solids.