when comparing similar 3d figures, there are relationships between the scale factor and the ratios of the surface areas as well as the volumes. Identify the correct relationships

The linear distances of similar 3D figures will have a relationship that is ??? the SF.

The surface areas of similar 3D figures will have a relationship that is ??? the SF.

The volumes of similar 3D figures will have a relationship that is ??? the SF.

1 answer

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:

  1. 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 \( k : 1 \).
  2. 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.
  3. 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