What is the ratio of the areas of the two triangles How does this ratio relate to the scale factor?

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What is a Scale Factor?
Watch How to use Scale Factors with Area and Volume on the DoingMaths YouTube channel
Enlarging With a Scale Factor of 5
Scale Factors With Area
Enlarging an Area by a Scale Factor
Enlarging a Volume by a Scale Factor
Questions & Answers

Suppose you have two similar figures , one larger than the other. The scale factor is the ratio of the length of a side of one figure to the length of the corresponding side of the other figure.

Example:

Here, X Y U V = 12 3 = 4 . So, the scale factor is 4 .

Note that when a two-dimensional figure is enlarged ( dilated ) by a scale factor of k , the area of the figure is changed by a factor of k 2 . In case of three-dimensional figures, when a figure is dilated by a scale factor of k , the volume is changed by a factor of k 3 .

I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.

What is a scale factor?

What is a Scale Factor?

When enlarging a shape or image, we use a scale factor to tell us how many times bigger we want each line/side to become. For example, if we enlarged a rectangle by scale factor 2, each side would become twice as long. If we enlarged by a scale factor of 10, each side would become 10 times as long.

The same idea works with fractional scale factors. A scale factor of 1/2 would make every side 1/2 as big (this is still called an enlargement, even though we have ended up with a smaller shape).

Watch How to use Scale Factors with Area and Volume on the DoingMaths YouTube channel

In this picture, the shape has been scaled by a factor of 5

Enlarging With a Scale Factor of 5

In the diagram above, the left-hand triangle has been enlarged by a scale factor of 5 to produce the triangle on the right. As you can see, each of the three side lengths of the original triangle has been multiplied by 5 to produce the side lengths of the new triangle.

Scale Factors With Area

But what effect does enlarging by a scale factor have on the area of a shape? Is the area also multiplied by the scale factor?

Let's look at an example.

Enlarging an area by scale factor

Enlarging an Area by a Scale Factor

In the diagram above, we have started with a rectangle of 3cm by 5cm and then enlarged this by a scale factor of 2 to get a new rectangle of 6cm by 10cm (each side has been multiplied by 2).

Look at what has happened to the areas:

Original area = 3 x 5 = 15cm2

New area = 6 x 10 = 60cm2

The new area is 4 times the size of the old area. By looking at the numbers we can see why this has happened.

The length and the height of the rectangle have both been multiplied by 2, hence when we find the area of the new rectangle we now have two lots of x2 in there, hence the area has been multiplied by 2 twice, the equivalent of multiplying by 4.

More formally, we can think of it like this:

After an enlargement of scale factor n:

New area = n x original length x n x original height

= n x n x original length x original height

= n2 x original area.

So to find the new area of an enlarged shape, you multiply the old area by the square of the scale factor.

This is true for all 2-d shapes, not just rectangles. The reasoning is the same; area is always two dimensions multiplied together. These dimensions are both being multiplied by the same scale factor, hence the area is multiplied by the scale factor squared.

Enlarging a volume by a scale factor

Enlarging a Volume by a Scale Factor

What about if we enlarge a volume by a scale factor?

Look at the diagram above. We have enlarged the left-hand cuboid by a scale factor of 3 to produce the cuboid on the right. You can see that each side has been multiplied by 3.

The volume of a cuboid is height x width x length, so:

Original volume = 2 x 3 x 6 = 36cm3

New volume = 9 x 6 x 18 = 972cm3

By using division we can quickly see that the new volume is actually 27 times larger than the original volume. But why is this?

When enlarging the area we needed to take into account how two multiplied sides were both being multiplied by the scale factor, hence we ended up using the square of the scale factor to find the new area.

For volume it is a very similar idea, however this time we have three dimensions to take into consideration. Again, each of these is being multiplied by the scale factor, so we need to multiply our original volume by the scale factor cubed.

More formally, we can think of it like this:

After an enlargement of scale factor n:

New volume = n x original length x n x original height x n x original width

= n x n x n x original length x original height x original width

= n3 x original volume.

So to find the new volume of an enlarged 3d shape, you multiply the old volume by the cube of the scale factor.

In summary, the rules of enlarging areas and volumes are very easy to remember, especially if you remember how we worked them out.

If you are enlarging by a scale factor n:

Enlarged length = n x original length

Enlarged area = n2 x original area

Enlarged volume = n3 x original volume.

Questions & Answers

Question: If you have 2 areas in a ratio, how do we find scale factors?

Answer: This works in a similar way to finding the scale factors for length and area. If you have a ratio for the areas of two similar shapes, then the ratio of the lengths would be the square roots of this area ratio. E.g. if the areas were in the ratio 3:5, the lengths would be in the ratio _/3:_/5. To get a scale factor from this we simplify the ratio into the form 1:n (in this case 1:_/(5/3)) and the right-hand side gives you the scale factor.

Three methods are shown below, but all of them use this idea. If you draw lines from the centre of the hexagon to each of its vertices, 6 identical isosceles triangles are formed (with the green angles equal). The blue angle is 360$^\circ\div$6 = 60$^\circ$, which means that the green angles must also be 60$^\circ$, and so the hexagon is split into 6 equilateral triangles - which have side length 1, the same as the hexagon.

Splitting both shapes into equilateral triangles with side length 1

We have seen that we can split the hexagon into 6 equilateral triangles with side length 1. We can try to do the same to the triangle.

Joining the midpoints of the sides, as shown on the right, we can split the triangle into 4 equilateral triangles with side length 2.

Doing the same to each of the smaller triangles formed gives equilateral triangles with side length 1. There are 4$\times$4 = 16 of these in the large equilateral triangle. So the hexagon contains 6 equilateral triangles of side length 1, whilst the triangle contains 16. So their areas are in the ratio 6:16, which simplifies to 3:8.

Scaling the hexagon up

6 copies of the triangle will fit together to make a hexagon with side length 4. So 6 copies of the triangle will make a copy of the hexagon which has been enlarged by a scale factor of 4. That means its area has been enlarged by a scale factor of 4$^2$ = 16. So 6 copies of the triangle have the same area as 16 copies of the hexagon.

So the ratio of the area of the hexagon to the area of the triangle is 6:16, which simplifies to 3:8

Scaling the triangle down

6 equilateral triangles of side length 1 fit into the hexagon. These triangles are similar to the equilateral triangle of side length 4, but their sides are 4 times shorter. This means that their area is 4$^2$ = 16 times smaller. So, when scaled down by a factor of 16, the area of the triangle fits 6 times into the area of the hexagon. So the area of the hexagon is $\frac6{16}=\frac38$ of the area of the triangle. So the ratio of the area of the hexagon to the area of the triangle is $\frac38$:1, which simplifies to 3:8.