When comparing similar 3D figures, the relationships between the scale factor (SF) and the ratios of linear distances, surface areas, and volumes can be expressed as follows:
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The linear distances of similar 3D figures will have a relationship that is equal to the SF.
- If the scale factor between two similar figures is \( k \), then all linear dimensions (lengths, widths, heights) have a ratio of \( k : 1 \).
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The surface areas of similar 3D figures will have a relationship that is the square of the SF.
- The ratio of the surface areas will be \( k^2 : 1 \), where \( k \) is the scale factor.
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The volumes of similar 3D figures will have a relationship that is the cube of the SF.
- The ratio of the volumes will be \( k^3 : 1 \), where \( k \) is the scale factor.
In summary:
- Linear distances: equal to the SF
- Surface areas: square of the SF
- Volumes: cube of the SF