Two soft toys are similar. One toy is three times the height of the other. How much more material is required to make it? Similarity:Volume and surface area of similar solids

Expert Answers

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Since the toys are similar the bigger toy does not only have 3 times greater height but also 3 times greater depth and width (if the toys are 3-dimensional). So the volume of bigger toy is 3*3*3=27 times greater. If the toys are 2-dimensional you will get that bigger toy has 3*3=9 times greater surface area.

In mathematics this is called homothetic transformation and ratio k which is in your case k=3. So for similar 2-dimensional objects with ratio k ratio of their surface area is `k^2` and for 3-dimensional objects ratio of their volumes is `k^3.`

For further explenation see e.g. Bruce E. Meserve: Fundamental Concepts of Geometry or one of the links below.

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