Ratio of Proportionality in Similar Figures

Similar Figures and Ratios of Proportionality

Introduction

  • When dealing with similar figures, the ratio of the areas is not the same as the ratio of proportionality; instead, it is the square of that ratio.
  • In three-dimensional figures, like cubes, similar principles apply but extend to volumes as well.

Cube Example

  • Consider two cubes to illustrate the concept.
  • Cube 1: 2x2x2
    • Volume: 2×2×2=82 \times 2 \times 2 = 8
  • Cube 2: 3x3x3
    • Volume: 3×3×3=273 \times 3 \times 3 = 27
  • These cubes are similar because they maintain the same shape but differ in size.

Ratio of Proportionality and Volumes

  • Ratio of Proportionality (small to big): 23\frac{2}{3}
  • Ratio of Volumes: 827\frac{8}{27}
  • It's important to note that 23≠827\frac{2}{3} \neq \frac{8}{27}
  • The relationship between the ratios is that the ratio of the volumes is the cube of the ratio of proportionality:
    • (23)3=2333=827\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}

Surface Area Analysis

  • Cube 1:
    • Area of one face: 2×2=42 \times 2 = 4
    • Surface area (6 faces): 6×4=246 \times 4 = 24
  • Cube 2:
    • Area of one face: 3×3=93 \times 3 = 9
    • Surface area (6 faces): 6×9=546 \times 9 = 54

Ratio of Surface Areas

  • Ratio of Surface Areas: 2454\frac{24}{54}
    • This fraction can be reduced by dividing both the numerator and the denominator by 6:
      • 24÷654÷6=49\frac{24 \div 6}{54 \div 6} = \frac{4}{9}
  • The ratio of the surface areas (49\frac{4}{9}) is not the same as the ratio of proportionality (23\frac{2}{3}).
  • However, the ratio of surface areas is the square of the ratio of proportionality:
    • (23)2=2232=49\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}

General Principles for Similar Solids

  • For two similar solids, if the ratio of proportionality is a:b, then:
    • Ratio of Areas: (ab)2(\frac{a}{b})^2
    • Ratio of Volumes: (ab)3(\frac{a}{b})^3

Real-World Example: Grapefruit Calories

  • Scenario: A 2-inch grapefruit contains 100 calories.
  • A grapefruit that is twice as big (4 inches) is purchased.
  • Common Mistake: Assuming that since the grapefruit is twice as big, it has 200 calories.
  • Correct Calculation:
    • The ratio of the radii is 1:2.
    • The ratio of the volumes is 13:23=1:81^3 : 2^3 = 1:8.
    • Therefore, the larger grapefruit has 100×8=800100 \times 8 = 800 calories.

Conclusion

  • It is crucial to remember that when scaling from lengths to areas to volumes, the ratios are not the same. The ratio of volumes is the cube of the ratio of the sides.