Suppose we have two shapes that look very similar, but one looks bigger than the other. We measure the lengths and indeed find that the lengths of the bigger shape are all exactly three times the lengths of the smaller shape. We then draw another shape, with sides five times the length of the smaller shape. There is a special name for this: the shapes are mathematically similar with a scale factor of three and five respectively! Luckily, in this article, we will be exploring everything that you need to know about similarity and in particular, scale factors. So, before we begin, let's start by defining some key terms.
Scale Factors Definition
Two similar triangles with scale factor 2- StudySmarter Originals
In the above image, we have two triangles. Notice that the lengths of the triangle 
are all exactly twice the lengths of the triangle 
. Other than that, the triangles are exactly the same. Therefore, we can say that the two shapes are similar with a scale factor of two. We can also say that the side 
corresponds to the side 
, the side 
corresponds to the side 
and the side 
corresponds to the side 
.
A scale factor tells us the factor by which a shape has been enlarged by. The corresponding sides are the sides of the shape that have proportional lengths.
If we have a shape enlarged by a scale factor of three, then each side of the shape is multiplied by three to produce the new shape.
Scale Factors Formulas
There is a very simple formula for working out the scale factor when we have two similar shapes. First, we need to identify the corresponding sides. Recall from earlier that these are the sides that are in proportion with each other. We then need to establish which is the original shape and which is the transformed shape. In other words, which is the shape that has been enlarged? This is usually stated in the question.
Then, we take an example of corresponding sides where the lengths of the sides are known and divide the length of the enlarged side by the length of the original side. This number is the scale factor.
Putting this mathematically, we have:
Where 
denotes the scale factor, 
denotes the enlarged figure side length and 
denotes the original figure side length and the side lengths taken are both from corresponding sides.
Scale Factors Examples
In this section, we will look at some further scale factors examples.
In the below image there are similar shapes 
and 
. We have:
, 
, 
, 
, 
and 
.
Work out the value of 
and 
.
Example working out missing lengths using scale factor - StudySmarter Originals
Solution:
Looking at the image, we can see that 
and 
are corresponding sides meaning that their lengths are in proportion with one another. Since we have the lengths of the two sides given, we can use this to work out the scale factor.
Calculating the scale factor, we have 
.
Thus, if we define 
to be the original shape, we can say that we can enlarge this shape with a scale factor of 
to produce the enlarged shape 
.
Now, to work out 
, we need to work backwards. We know that 
and 
are corresponding sides. Thus, to get from 
to 
we must divide by the scale factor. We can say that 
.
To work out y, we need to multiply the length of the side 
by the scale factor. Thus, we have 
.
Therefore 
and 
.
Below are three similar quadrilaterals. We have that 
, 
, 
and 
. Work out the area of quadrilaterals 
and 
.
Example working out the area using scale factor - StudySmarter Originals
Solution:
First, let's work out the scale factor to get from 
to 
. Since 
and 
, we can say that the scale factor 
. Thus, to get from 
to 
we enlarge by a scale factor of 
. We can therefore say that the length of 
is 
.
Now, let's work out the scale factor to get from 
to 
. Since 
and 
, we can say that the scale factor 
. Thus, to work out A''D'', we multiply the length of A'D' by 
to get 
.
To work out the area of a quadrilateral, recall that we multiply the base by the height. So, the area of 
is 
and similarly, the area of 
is 
.
Below are two similar right-angled triangles 
and 
. Work out the length of 
.
Working out missing length using scale factor and pythagoras - StudySmarter Originals
Solution:
As usual, let's start by working out the scale factor. Notice that 
and 
are two known corresponding sides so we can use them to work out the scale factor.
So, 
. Thus, the scale factor is 
. Since we do not know the side 
, we cannot use the scale factor to work out 
. However, since we know 
, we can use it to work out 
.
Doing so, we have 
. Now we have two sides of a right-angled triangle. You may remember learning about Pythagoras' theorem. If not, perhaps review this first before continuing with this example. However, if you are familiar with Pythagoras, can you work out what we need to do now?
According to Pythagoras himself, we have that 
where
is the hypotenuse of a right-angled triangle, and 
and 
are the other two sides. If we define 
, 
, and 
, we can use Pythagoras to work out 
!
Doing so, we get 
. So, 
.
We therefore have that 
.
Scale Factor Enlargement
If we have a shape and a scale factor, we can enlarge a shape to produce a transformation of the original shape. This is called an enlargement transformation. In this section, we will be looking at some examples relating to enlargement transformations.
There are a few steps involved when enlarging a shape. We first need to know how much we are enlarging the shape which is indicated by the scale factor. We also need to know where exactly we are enlarging the shape. This is indicated by the centre of enlargement.
The centre of enlargement is the coordinate that indicates where to enlarge a shape.
We use the centre of enlargement by looking at a point of the original shape and working out how far it is from the centre of enlargement. If the scale factor is two, we want the transformed shape to be twice as far from the centre of enlargement as the original shape.
We will now look at some examples to help understand the steps involved in enlarging a shape.
Below is triangle 
. Enlarge this triangle with a scale factor of 
with the centre of enlargement at the origin.
Example of enlarging a triangle - StudySmarter Originals
Solution:
The first step in doing this is to make sure the centre of enlargement is labelled. Recall that the origin is the coordinate 
. As we can see in the above image, this has been marked in as point O.
Now, pick a point on the shape. Below, I have chosen point B. To get from the centre of enlargement O to point B, we need to travel 
unit along and 
unit up. If we want to enlarge this with a scale factor of 
, we will need to travel 
units along and 
units up from the centre of enlargement. Thus, the new point 
is at the point 
.
Example of enlarging a triangle - StudySmarter Originals
We can now label the point 
on our diagram as shown below.
Example of enlarging a triangle point by point - StudySmarter Originals
Next, we do the same with another point. I have chosen 
. To get from the centre of enlargement O to point C, we need to travel 
units along and 
unit up. If we enlarge this by 
, we will need to travel 
units along and 
units up. Thus, the new point 
is at 
.
Example of enlarging a triangle point by point - StudySmarter Originals
We can now label the point 
on our diagram as shown below.
Example of enlarging a triangle point by point - StudySmarter Originals
FInally, we look at the point 
. To get from the centre of enlargement O to the point A, we travel 
unit along and 
units up. Thus, if we enlarge this by a scale factor of 
, we will need to travel 
units along and 
units up. Therefore, the new point 
will be at the point 
.
Example of enlarging a triangle point by point - StudySmarter Originals
We can now label the point 
on our diagram as shown below. If we join up the coordinates of the points we have added, we end up with the triangle 
. This is identical to the original triangle, the sides are just three times as big. It is in the correct place as we have enlarged it relative to the centre of enlargement.
Example of enlarging a triangle - StudySmarter Originals
Therefore, we have our final triangle depicted below.
Example of enlarging a triangle - StudySmarter Originals
Negative Scale Factors
So far, we have only looked at positive scale factors. We have also seen some examples involving fractional scale factors. However, we can also have negative scale factors when transforming shapes. In terms of the actual enlargement, the only thing that really changes is that the shape appears to be upside down in a different position. We will see this in the below example.
Below is quadrilateral 
. Enlarge this quadrilateral with a scale factor of 
with the centre of enlargement at the point 
.
Negative scale factors example - StudySmarter Originals
Solution:
First, we take a point on the quadrilateral. I have chosen point 
. Now, we need to work out how far D is from the centre of enlargement P. In this case, to travel from P to D, we need to travel 
unit along and 
unit up.
If we want to enlarge this with a scale factor of 
, we need to travel 
units along and 
units up. In other words, we are moving 
units away and 
units down from P. The new point D' is therefore at 
, as shown below.
Negative scale factors example - StudySmarter Originals
Now, consider point A. To get from P to A, we travel 
unit along and 
units up. Therefore, to enlarge this with a scale factor 
, we travel 
units along and 
units up. In other words, we travel 
units to the left of P and 
units down, as shown as point A' below.
Negative scale factors example - StudySmarter Originals
Now, consider point C. To get from P to C, we travel 
units along and 
unit up. Therefore, to enlarge this with a scale factor 
, we travel 
units along and 
units up. In other words, we travel 
units to the left of P and 
units down, as shown as point C' below.
Negative scale factors example - StudySmarter Originals
Now, consider point B. To get from P to B, we travel 
units along and 
units up. Therefore, to enlarge this with a scale factor 
, we travel 
units along and 
units up. In other words, we travel 
units to the left of P and 
units down, as shown as point B' below.
Negative scale factors example - StudySmarter Originals
If we join up the points, and remove the ray lines, we obtain the below quadrilateral. This is our final enlarged shape. Notice that the new image appears upside down.
Negative scale factors example - StudySmarter Originals
Scale Factors - Key takeaways
- A scale factor tells us the factor by which a shape has been enlarged by.
- For example, if we have a shape enlarged by a scale factor of three, then each side of the shape is multiplied by three to produce the new shape.
- The corresponding sides are the sides of the shape that have proportional lengths.
- If we have a shape and a scale factor, we can enlarge a shape to produce a transformation of the original shape. This is called an enlargement transformation.
- The centre of enlargement is the coordinate that indicates where to enlarge a shape.
- We can also have negative scale factors when transforming shapes. In terms of the actual enlargement, the shape will just appear to be upside down.
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and 
. By looking at the shapes, we can see that 
corresponds with 
because they are both nearly identical- the only difference is 
is longer. By how much?
is two 
is six 
by the length of 
. Thus, the scale factor is
.
and the corresponding sides are 
with 
, 
with 
, 
with 
and 
with 
. 
and 
, both drawn to scale. Work out the scale factor to get from 
to 
. 
and 
for example. We then divide the length of the transformed side by the length of the original side. In this case, 
and 
. 
. 
