Scale factor of length, surface area and volume

Scale Factor of Length

The scale factor is the ratio of the lengths of corresponding sides in two similar shapes.
If the scale factor of length is k, each dimension in the new shape is k times larger than the corresponding dimension in the original shape.
When scaling down, the scale factor will be between 0 and 1. When scaling up, the value of k will be greater than 1.

Scale Factor of Surface Area

The scale factor of surface area is the scale factor of length squared.
If the scale factor of length is k, the surface area of the new shape is k² times larger than the surface area of the original shape.
This stems from the fact that area is a 2-dimensional measurement, hence the scale factor must be squared.

Scale Factor of Volume

The scale factor of volume is the scale factor of length cubed.
If the scale factor of length is k, the volume of the new shape is k³ times larger than the volume of the original shape.
This is because volume is a 3-dimensional measurement, thus the scale factor must be cubed.

Practical Applications

Understanding scale factors can be particularly useful in a range of real-world scenarios, including map reading, architectural designing, model building, and more.
Applying correct scale factors ensures that all aspects of an item keep the same proportions, preserving the authenticity of the original item in the scaled version.

Ensure that you comprehend these fundamental principles and regularly practice problems related to scale factors to cement your understanding. In particular, remember to square the scale factor when dealing with area, and cube it when working with volume.