6th Grade Math Textbook, Fundamentals

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Enrichment:

Three-Dimensional Figures and the Ratio of Similarity

Objective To explore the relationship between the ratio of similarity and the ratios
of the area and volumes of similar solids

You know that if two 2-dimensional figures are
similar, then the ratio of their areas is equal to the
square of the ratio of similarity.

For example, triangle ABCand triangle DEF are similar.
The ratios of the lengths of the corresponding sides
is 2 : 1, so the ratio of similarity is 2 : 1.

The area of triangle ABCis (8)(6) 24 square units.

The area of triangle DEFis (4)(3) 6 square units.

The ratio of the areas is 24 : 6, which is equivalent to 4 : 1; and 4 : 1 is
equivalent to 2^2 : 1^2. So the ratio of the areas of the triangles is the
square of the ratio of similarity.

Now you will explore the ratio of similarity and the

surface areas and volumes of similar 3-dimensional

figures.

The ratio of similarity for similar solids is the ratio of

the corresponding linear parts or linear segments.

Use dot paper to draw two similar rectangular prisms.

You may use dimensions other than those in the drawings below.

Triangle DEF

Side Length

DE 3 units

FD 5 units

EF 4 units

Triangle ABC

Side Length

AB 6 units

CA 10 units

BC 8 units

1

2

1

2

BC

F

A

D

E

Face

B

Face A

Prism 1

Face

C Face A‘

Prism 2

Face B‘

Face

C‘