Scale Factors


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 Scale Factors

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    Scale Factors Definition

    Scale factors definition- two similar triangles with scale factor 2.Two similar triangles with scale factor 2- StudySmarter Originals

    In the above image, we have two triangles. Notice that the lengths of the triangle A'B'C' are all exactly twice the lengths of the triangle ABC. 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 AB corresponds to the side A'B', the side AC corresponds to the side A'C' and the side BC corresponds to the side B'C'.

    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.

    Below is another example of a set of similar shapes. Can you work out the scale factor and corresponding sides?

    scale factor definition- working out the scale factor for two quadrilaterals ABCD and A'B'C'D'.Working out scale factor example with quadrilaterals - StudySmarter Originals

    Solution:

    We have two quadrilaterals ABCD and A'B'C'D'. By looking at the shapes, we can see that BC corresponds with B'C' because they are both nearly identical- the only difference is B'C' is longer. By how much?

    Counting the squares, we can see that BC is two units long, and B'C' is six units long. To work out the scale factor, we divide the length of BC by the length of B'C'. Thus, the scale factor is62=3 .

    We can conclude that the scale factor is 3 and the corresponding sides are AB with A'B', BC with B'C', CD with C'D' and AD with A'D'.

    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:

    SF= ab

    Where SF denotes the scale factor, a denotes the enlarged figure side length and b 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 ABCDE and A'B'C'D'E'. We have:

    DC=16 cm, D'C'=64 cm , ED= x cm, E'D'=32 cm, AB=4 cm and A'B'=y cm.

    AB=4 cmWork out the value of x and y.

    scale factors examples- example working out missing lengths using scale factorExample working out missing lengths using scale factor - StudySmarter Originals

    Solution:

    Looking at the image, we can see that DC and D'C' 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 SF=6416=4.

    Thus, if we define ABCDE to be the original shape, we can say that we can enlarge this shape with a scale factor of 4 to produce the enlarged shape A'B'C'D'E'.

    Now, to work out x, we need to work backwards. We know that ED and E'D' are corresponding sides. Thus, to get from E'D' to ED we must divide by the scale factor. We can say that x=324=8 cm .

    To work out y, we need to multiply the length of the side AB by the scale factor. Thus, we have A'B'=4×4=16 cm.

    Therefore x=8 cm and y=16 cm.

    Below are similar triangles ABC and A'B'C', both drawn to scale. Work out the scale factor to get from ABC to A'B'C'.

    scale factors examples- example working out fractional scale factorExample working out the scale factor where scale factor is fractional - StudySmarter Originals

    Solution:

    Notice in this shape, the transformed shape is smaller than the original shape. However, to work out the scale factor, we do the exact same thing. We look at two corresponding sides, let's take AB and A'B' for example. We then divide the length of the transformed side by the length of the original side. In this case, AB= 4 units and A'B'= 2 units.

    Therefore, the scale factor, SF=24=12 .

    Notice here that we have a fractional scale factor. This is always the case when we go from a bigger shape to a smaller shape.

    Below are three similar quadrilaterals. We have that DC=10 cm, D'C'=15 cm, D''C''= 20 cm and A'D'= 18 cm . Work out the area of quadrilaterals ABCDand A''B''C''D''.

    scale factors examples- example working out area using scale factorExample working out the area using scale factor - StudySmarter Originals

    Solution:

    First, let's work out the scale factor to get from ABCD to A'B'C'D'. Since D'C'=15 cm and DC= 10 cm, we can say that the scale factor SF=1510=1.5 . Thus, to get from ABCD to A'B'C'D' we enlarge by a scale factor of 1.5. We can therefore say that the length of AD is 181.5=12 cm.

    Now, let's work out the scale factor to get from A'B'C'D' to A''B''C''D''. Since D''C''=20 cm and D'C'=15 cm, we can say that the scale factor SF=2015=43. Thus, to work out A''D'', we multiply the length of A'D' by 43 to get A''D''=18×43=24 cm.

    To work out the area of a quadrilateral, recall that we multiply the base by the height. So, the area of ABCD is 10 cm×12 cm=120 cm2 and similarly, the area of A''B''C''D'' is 20 cm ×24 cm= 420 cm2.

    Below are two similar right-angled triangles ABC and A'B'C'. Work out the length of A'C'.

    scale factors examples- example involving pythagorasWorking out missing length using scale factor and pythagoras - StudySmarter Originals

    Solution:

    As usual, let's start by working out the scale factor. Notice that BC and B'C' are two known corresponding sides so we can use them to work out the scale factor.

    So, SF= 42=2 . Thus, the scale factor is 2. Since we do not know the side AC, we cannot use the scale factor to work out A'C'. However, since we know AB, we can use it to work out A'B'.

    Doing so, we have A'B'= 3 × 2=6 cm. 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 a2+b2=c2wherec is the hypotenuse of a right-angled triangle, and a and b are the other two sides. If we define a=4 cm, b=6 cm, and c=A'C', we can use Pythagoras to work out c!

    Doing so, we get c2=42+62=16+36=52. So, c=52=7.21 cm.

    We therefore have that A'C'=7.21 cm.

    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 ABC. Enlarge this triangle with a scale factor of 3 with the centre of enlargement at the origin.

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample 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 (0,0). 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 1 unit along and 1 unit up. If we want to enlarge this with a scale factor of 3, we will need to travel 3 units along and 3 units up from the centre of enlargement. Thus, the new point B' is at the point (3,3).

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle - StudySmarter Originals

    We can now label the point B' on our diagram as shown below.

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle point by point - StudySmarter Originals

    Next, we do the same with another point. I have chosen C. To get from the centre of enlargement O to point C, we need to travel 3 units along and 1 unit up. If we enlarge this by 3, we will need to travel 3×3=9 units along and 1×3=3 units up. Thus, the new point C' is at (9,3).

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle point by point - StudySmarter Originals

    We can now label the point C' on our diagram as shown below.

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle point by point - StudySmarter Originals

    FInally, we look at the point A. To get from the centre of enlargement O to the point A, we travel 1 unit along and 4 units up. Thus, if we enlarge this by a scale factor of 3, we will need to travel 1×3=3 units along and 4×3=12 units up. Therefore, the new point A' will be at the point (3,12).

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle point by point - StudySmarter Originals

    We can now label the point A' on our diagram as shown below. If we join up the coordinates of the points we have added, we end up with the triangle A'B'C'. 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.

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample of enlarging a triangle - StudySmarter Originals

    Therefore, we have our final triangle depicted below.

    scale factor enlargement- example with triangle being enlarged with scale factor 3 and centre of enlargement at the originExample 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 ABCD. Enlarge this quadrilateral with a scale factor of -2 with the centre of enlargement at the point P=(1,1).

    negative scale factors- example with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)Negative scale factors example - StudySmarter Originals

    Solution:

    First, we take a point on the quadrilateral. I have chosen point D. 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 1 unit along and 1 unit up.

    If we want to enlarge this with a scale factor of -2, we need to travel 1×-2=-2 units along and 1×-2=-2 units up. In other words, we are moving 2 units away and 2 units down from P. The new point D' is therefore at (-1,-1), as shown below.

    negative scale factors- example with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)Negative scale factors example - StudySmarter Originals

    Now, consider point A. To get from P to A, we travel 1 unit along and 2 units up. Therefore, to enlarge this with a scale factor -2, we travel 1×-2=-2 units along and 2×-2=-4 units up. In other words, we travel 2 units to the left of P and 4 units down, as shown as point A' below.

    negative scale factors- example with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)Negative scale factors example - StudySmarter Originals

    Now, consider point C. To get from P to C, we travel 3 units along and 1 unit up. Therefore, to enlarge this with a scale factor -2, we travel 3×-2=-6 units along and 1×-2=-2 units up. In other words, we travel 6 units to the left of P and 2 units down, as shown as point C' below.

    negative scale factors- example with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)Negative scale factors example - StudySmarter Originals

    Now, consider point B. To get from P to B, we travel 2 units along and 2 units up. Therefore, to enlarge this with a scale factor -2, we travel 2×-2=-4 units along and 2×-2=-4 units up. In other words, we travel 4 units to the left of P and 4 units down, as shown as point B' below.

    negative scale factors- example with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)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 with quadrilater being enlarged with scale factor -2 and centre of enlargement (1,1)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.
    Frequently Asked Questions about Scale Factors

    What is a scale factor?

    When we enlarge a shape, the scale factor is the quantity by which each side is enlarged by. 

    What is a scale factor of 3?

    When we enlarge a shape, we enlarge it by a scale factor of three when we multiply each of the sides by three to get the new shape. 

    How do you find the missing length of a scale factor?

    If we know the scale factor, we can multiply the side of the original shape by the scale factor to find the missing lengths of the new shape. Alternatively, if we have known sides of the enlarged shapes, we can divide the lengths by the scale factor to get the lengths of the original shape. 

    How do you find the scale factor of an enlargement?

    Divide the corresponding sides of the enlarged shape by the original shape.

    What happens if a scale factor is negative?

    The shape is turned upside down.

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