# Length, Area and Volume scale factors of mathematically similar shapes.

On this page you will learn how two calculate the area or volume of two mathematically similar shapes. Similar shapes are shapes whose lengths have been enlarged by the same scale factor.

To solve these type of problems first calculate the length scale factor (do this by dividing two corresponding sides). If you are finding an area you will need to square the length scale factor, or if you are finding a volume (or mass) you will need to cube the length scale factor. You can then multiply or divide the area or volume by the scale factor to work out the required area or volume.

**Example**

Here are two mathematically similar boxes. Work out the surface area of the larger box if the smaller box has a surface area of 40cm².

First you need to work out the length scale factor. Do this by dividing the two corresponding side lengths of the boxes:

12 ÷ 6 = 2

Since you are finding an area you will need to square the length scale factor to give the area scale factor:

2² = 4

So the surface area of the larger box is 4 times the size of the smaller box.

Finally, multiply the surface area of the smaller box by 4 to give the surface area of the larger box:

40 × 4 = 160cm²

A common mistake with this question is to forget about working out the scale factor and just multiply the area of the small box by 2.

**Example 2**

Amy has two mathematically similar bottles of perfume. The radius of the small bottle of perfume is 2cm and 5cm on the larger bottle. If the small bottle of perfume contains 20ml of perfume work out the volume of the large bottle of perfume.

As the last example begin by finding the length scale factor:

5 ÷ 2 = 2.5

Since you are doing a volume problem you will need the volume scale factor. You can get the volume scale factor by cubing the length scale factor:

2.5³ = 15.625

So the volume of the larger bottle of perfume is 15.625 times bigger than the smaller bottle (not 2.5!)

All you need to do now is multiply the volume of the small bottle of perfume by 15.625:

20 × 15.625 = 312.5ml

So to summarise, always remember to square the length scale factor if you are working on an area problem or cube the length scale factor if you are doing a volume problem.

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