To find the scale factor between two similar solids, you need to compare the lengths of corresponding sides.
Here's a step-by-step breakdown:
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Identify Corresponding Sides: Determine which sides of the two solids are similar. For instance, if you're comparing two cylinders, the heights correspond to each other, and the radii correspond to each other. For polyhedra, make sure to identify which edges on the smaller and larger shapes match.
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Form a Ratio: Create a ratio by dividing the length of a side of one solid by the length of the corresponding side of the other solid. Ensure you are consistent with which solid is in the numerator and which is in the denominator.
Scale Factor = (Length of side on Solid A) / (Length of corresponding side on Solid B)
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Simplify the Ratio: Simplify the resulting fraction. This simplified ratio represents the scale factor.
Example:
Let's say you have two similar rectangular prisms.
- Prism A has a length of 6 cm, a width of 3 cm, and a height of 2 cm.
- Prism B has a length of 12 cm, a width of 6 cm, and a height of 4 cm.
To find the scale factor:
- Using the lengths: Scale Factor = 12 cm / 6 cm = 2
- Using the widths: Scale Factor = 6 cm / 3 cm = 2
- Using the heights: Scale Factor = 4 cm / 2 cm = 2
In this case, the scale factor is 2. This means that Prism B is twice as large as Prism A. We could also say that Prism A is half the size of Prism B, in which case the scale factor would be 1/2 or 0.5. Which solid you choose to be 'Solid A' and 'Solid B' affects whether the scale factor is greater than one (enlargement) or less than one (reduction).
Important Notes:
- The solids must be similar for a single scale factor to exist. If the ratios of corresponding sides are not equal, then the solids are not similar, and there is no single scale factor.
- The scale factor applies to all linear dimensions (length, width, height, radius, etc.).
- Once you have the scale factor for the sides (a linear scale factor), you can determine the scale factor for the surface areas (square of the linear scale factor) and volumes (cube of the linear scale factor).
