Area Scale Factor

This chapter explores area scale factors. You’ll discover the connection between the linear scale factor and the area scale factor of similar shapes. You’ll also explore working with lengths and areas of similar shapes. To understand area scale factors well it is necessary to use the following example. Below are two similar squares one smaller while the other bigger.
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You might want to try and guess how many small squares can be fit in the large square. The answer is not that obvious and with areas it is less obvious than lengths. Let us try to draw the squares in the large square.
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We can see that 2 cm length fit into the large 6cm length square. We using can conclude that;
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Here is another example except using rectangles.
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Four rectangles of the small rectangle shown can fit into the large rectangle. Above we can see that.
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Finding Scale Factors

In the above examples you might have noticed a pattern and connection between linear scale factors and area scale factors. To find the linear scale factor we simply divide the lengths of the biggest by the smallest and that goes for area scale factors; We divide the biggest area with the smallest as shown in the following example.

Finding missing areas

Below we have two similar triangles to demonstrate the linear and area scale factors.
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One of the areas above is unknown, there is a need for finding it. The first step is finding the linear scale factor since we’re able to find this; The area for finding the linear scale factor we know;
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This must mean;
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Next we find that area scale factor. In this case the area scale factor is;
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It is just the linear scale factor squared. That must mean;
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Since we know the area scale factor we can simply just use it to find the area for the largest shape by multiplying it with the smaller shape.
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Last example

Below are two footprints that are similar. Suppose we wanted to find it. This is an example of this common example;
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The small footprint has an area of 80cm² while the large footprint has an area of 500cm². The small footprint has a width of 6cm. Suppose we wanted to find the width of the large footprint. First we find the area scale factor as shown below;
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Next we find the linear scale factor, we know that;
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That must mean that;
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Therefore;
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The width is therefore;
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