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THE r:r²:r³ THEOREM

THE r:r²:r³ THEOREM. R. R³. R². R. When two figures are similar, the following is true. The ratio of their sides is the same as the ratio of their perimeters. RATIO: 3:4. Perimeter ₁: 3 + 3 + 6 + 6 = 18 Perimeter₂: 4 + 4 + 8 + 8 = 24 18:24 REDUCES TO 3:4.

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THE r:r²:r³ THEOREM

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  1. THE r:r²:r³ THEOREM R R³ R²

  2. R • When two figures are similar, the following is true. • The ratio of their sides is the same as the ratio of their perimeters. • RATIO: 3:4 Perimeter ₁: 3 + 3 + 6 + 6 = 18 Perimeter₂: 4 + 4 + 8 + 8 = 24 18:24 REDUCES TO 3:4 8 6 3 4

  3. R² • When two figures are similar, the following is true. • The ratio of their area is the ratio of their sides “squared”. • RATIO: 3:4 • RATIO²: 9:16 Area₁: (3 x 6) = 18 Area₂: (4 x 8) = 32 Ratio²: 18:32 REDUCES TO 9:16 8 6 3 4

  4. R³ • When two figures are similar, the following is true. • The ratio of their volume is the ratio of their sides “cubed”. • RATIO: 3:4 • RATIO³: 27:64 Volume₁: (3 x 6 x 6) = 108 Volume₂: (4 x 8 x 8) = 256 Ratio³: 108:256 REDUCES TO 27:64 8 6 3 4 6 8

  5. GUIDED PRACTICE Suppose the ratio of the sides of 2 cubes is 3:5. Then the ratios of their surface are is …. What is the ratio of their volume? RATIO of sides is 3:5 RATIO of surface area (3:5) ² = (9:25) RATIO of volume (3:5)³ = (27:125)

  6. APPLICATION The tetrahedra have a ratio of 3:5. If the volume of the small tetrahedron is 65 cubic units, then the volume of the large tetrahedron is . . . We are dealing with volume so we will use r³ (3/5)³ = 27/125 soooooo 65/Vlg = 27/125 cross multiply 125(65) = 27(Vlg) Divide 300.93 cubic units

  7. Similarity of Length, Area and Volume Two rectangular prisms are similar. Suppose the ratio of their vertical edges is 3:8. Use the r:r²:r³ Theorem to find the following without knowing the dimensions of the prism. • Find the ratio of their surface areas. • Find the ratio of their volumes. • The perimeter of the front face of the large prism is 18 units. Find the perimeter of the front face of the small prism. • The area of the front face of the large prism is 15 square units. Find the are of the front face of the small prism. • The volume of the small prism is 21 cubic units. Find the volume of the large prism.

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