# Area and Volume scale factor

A scale factor is a number which scales, or multiplies, some quantity. For example in this equation y=Cx, C is the scale factor for x. Scale factors are straight forward and are not as complex as other maths problems you might encounter. Once you understand the logic of solving them everything else becomes easy, there are just a few simple rules you have to understand. We shall be looking at scale factors to do with Areas and Volumes, but there are many others.

## Summary

Two figures/shapes may be identical in shape but generally not in size. The following shapes will always be similar in shape but not in size.

## Linear scale factor

A shape can be transformed into another similar shape by adjusting its size through enlargement/reduction by using a scale factor. In this process all dimensions of the shape must stay proportional to each other (i.e the lengths, heights, or widths)

Enlargement: In mathematics, an enlargement is a uniform scaling (transformation) that increases distances, areas and volumes.

For example a scale factor of 2 means that the new shape is twice the size of the original shape. A scale factor of 3 means that the new shape is three times the size of the original.

Reduction: Reduction is the opposite of enlargement where distances, areas, and volumes are decreased using a uniform scaling.

To carry out enlargements/reductions you must first find the scale factor. To find the enlargement scale factor you divide the dimensions of the big shape by the small shape.

To find the shape reduction scale factor you divide the small dimensions by the big dimensions.

Below is an example.

Example: The following shapes abcd and ABCD are similar. Find the length of CD.
Answer: Here we have to find the length of the bigger rectangle. We shall therefore be calculating an enlargement scale factor first.
Notice we have used the corresponding lengths. The rectangle ABCD is 1.5 times bigger than rectangle abcd. We therefore multiply the width of the smaller rectangle with the enlargement scale factor to find the size of the bigger width.

## Area Scale Factor

Area scale factor can apply to any shape provided you have at least one of the corresponding quantities (areas or lengths) For example here are two triangles; There are simple steps you will take to find the area of the large triangle;
Step 1:Find the linear/length scale factor.
In general to find the scale factor we divide the large quantity with the small quantity.
Step 2:Find the Area scale factor by squaring the linear scale factor
Step 3:Multiply the area scale factor with the small area to find the area for the large triangle.
The area for the large shape is 27cm2
To find one of the lengths given the areas you will need first find the area scale factors and then square root it to find the scale factor. You would then multiply if finding a length for a large shape or divide if finding the length for a small shape.

## Volume Scale Factor

The same strategy applies to volume scale factors, except you have to be aware that volumes are cubic roots. For example your might have the following two bottles whose lengths are known and just one volume, and you have to find the other volume.
Step 1: First step is to find the linear scale factor, remember that;
Step 2: Find the volume scale factor and then cubic it to find the volume scale factor.
Step 3Multiply the small volume with the volume scale factor
When working with scale factors always remember to first find the scale factor between the two shapes or quantities and and then multiply or divide the given quantity depending on whether you want to perform a reduction or an enlargement.
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A scale factor is a number which scales, or multiplies, some quantity. For example in this equation y=Cx, C is the scale factor for x. Scale factors are straight forward and are not as complex as other maths problems you might encounter. Once you understand the logic of solving them everything else becomes easy, there are just a few simple rules you have to understand. We shall be looking at scale factors to do with Areas and Volumes, but there are many others.

## Summary

Two figures/shapes may be identical in shape but generally not in size. The following shapes will always be similar in shape but not in size.

## Linear scale factor

A shape can be transformed into another similar shape by adjusting its size through enlargement/reduction by using a scale factor. In this process all dimensions of the shape must stay proportional to each other (i.e the lengths, heights, or widths)

Enlargement: In mathematics, an enlargement is a uniform scaling (transformation) that increases distances, areas and volumes.

For example a scale factor of 2 means that the new shape is twice the size of the original shape. A scale factor of 3 means that the new shape is three times the size of the original.

Reduction: Reduction is the opposite of enlargement where distances, areas, and volumes are decreased using a uniform scaling.

To carry out enlargements/reductions you must first find the scale factor. To find the enlargement scale factor you divide the dimensions of the big shape by the small shape.

To find the shape reduction scale factor you divide the small dimensions by the big dimensions.

Below is an example.

Example: The following shapes abcd and ABCD are similar. Find the length of CD.
Answer: Here we have to find the length of the bigger rectangle. We shall therefore be calculating an enlargement scale factor first.
Notice we have used the corresponding lengths. The rectangle ABCD is 1.5 times bigger than rectangle abcd. We therefore multiply the width of the smaller rectangle with the enlargement scale factor to find the size of the bigger width.

## Area Scale Factor

Area scale factor can apply to any shape provided you have at least one of the corresponding quantities (areas or lengths) For example here are two triangles; There are simple steps you will take to find the area of the large triangle;
Step 1:Find the linear/length scale factor.
In general to find the scale factor we divide the large quantity with the small quantity.
Step 2:Find the Area scale factor by squaring the linear scale factor
Step 3:Multiply the area scale factor with the small area to find the area for the large triangle.
The area for the large shape is 27cm2
To find one of the lengths given the areas you will need first find the area scale factors and then square root it to find the scale factor. You would then multiply if finding a length for a large shape or divide if finding the length for a small shape.

## Volume Scale Factor

The same strategy applies to volume scale factors, except you have to be aware that volumes are cubic roots. For example your might have the following two bottles whose lengths are known and just one volume, and you have to find the other volume.
Step 1: First step is to find the linear scale factor, remember that;
Step 2: Find the volume scale factor and then cubic it to find the volume scale factor.
Step 3Multiply the small volume with the volume scale factor
When working with scale factors always remember to first find the scale factor between the two shapes or quantities and and then multiply or divide the given quantity depending on whether you want to perform a reduction or an enlargement.
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