THE r:r²:r³ THEOREM

R

R²

R³

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

R

3 6

4

8

Perimeter ₁: 3 + 3 + 6 + 6 = 18

Perimeter₂: 4 + 4 + 8 + 8 = 24

18:24 REDUCES TO 3:4

When two figures are similar, the following is true.

The ratio of their area is the ratio of their sides “squared”.

RATIO: 3:4RATIO²: 9:16

R²

3 4

68

Area₁: (3 x 6) = 18

Area₂: (4 x 8) = 32

Ratio²:

18:32 REDUCES TO 9:16

When two figures are similar, the following is true.

The ratio of their volume is the ratio of their sides “cubed”.

RATIO: 3:4RATIO³: 27:64

R³

3

6

6

4

8

8

Volume₁: (3 x 6 x 6) = 108

Volume₂: (4 x 8 x 8) = 256

Ratio³:

108:256 REDUCES TO 27:64

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)

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 . . .

APPLICATION

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

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.

a) Find the ratio of their surface areas.b) Find the ratio of their volumes.c) The perimeter of the front face of the

large prism is 18 units. Find the perimeter of the front face of the small prism.

d) 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.

e) The volume of the small prism is 21 cubic units. Find the volume of the large prism.

Similarity of Length, Area and Volume