Similar Solids: Two solids are similar if they are the same type of solid and their corresponding dimensions such as radii, heights, base lengths, and widths are proportional.
Applications: Used in solving problems involving ratios between similar solids in geometry.
Key Concepts
Similarity in Solids
Definition: Solids with all corresponding angles congruent and corresponding sides proportional.
Examples: Similar triangular pyramids, prisms, cylinders, spheres.
Surface Areas of Similar Solids
Surface Area Ratio:
If two solids are similar with scale factor (\frac{a}{b}), their surface areas are in the ratio ((\frac{a}{b})^2).
Example: If scale factor is 2:3, then surface area ratio is ((\frac{2}{3})^2 = \frac{4}{9}).
Volumes of Similar Solids
Volume Ratio:
If two solids are similar with scale factor (\frac{a}{b}), their volumes are in the ratio ((\frac{a}{b})^3).
Example: If scale factor is 2:3, then volume ratio is ((\frac{2}{3})^3 = \frac{8}{27}).
Summary Table
Ratios
Units
Scale Factor
(\frac{a}{b})
in, ft, cm, m, etc.
Ratio of Surface Areas
((\frac{a}{b})^2)
in², ft², cm², m²
Ratio of Volumes
((\frac{a}{b})^3)
in³, ft³, cm³, m³
Examples
Example 1: Triangular Pyramids
Task: Determine if two triangular pyramids are similar.
Solution: Match corresponding parts. If any ratio is not equal, then they are not similar.
Example 2: Triangular Prisms
Task: Find missing sides in similar triangular prisms with given volume ratio 343:125.
Solution:
Determine scale factor by cube root: (\sqrt[3]{\frac{343}{125}} = \frac{7}{5}).
Use scale factor to find missing dimensions.
Example 3: Rectangular Prisms
Task: Check if two rectangular prisms are similar.
Solution: Check ratios of corresponding dimensions. If all are equal, they are similar.
Example 4: Cylinders
Task: Find height of taller cylinder with area ratio 16:25.