 When you change the size of a shape by making it larger or smaller, the degree to which you change its size is measured by something called its scale factor. A linear scale factor is simply the size of the enlargement or reduction of a shape. For instance, a scale factor of 2 means that the new shape is twice the size of the original. A scale factor of a half tells us the new shape is half the size of the original. To calculate the linear scale factor, we use the following formulae. Let's try an example of a linear scale factor. A, B, C, D, and A, B, C, D are similar shapes, not drawn to scale. If A, C equals 3, B, D equals 5, and A, C, 9, what is B, D? B, D is on the enlarged figure, so we will need to calculate an enlargement scale factor. We divide the size of the big known side, A, C, which is 9, and divide it by the size of the small known side, A, C, which is 3. The enlargement scale factor is therefore 3. To calculate the size of B, D, we multiply B, D by the scale factor of 3, and we get 15. The next scale factor we will take a look at is area scale factor. Suppose we have a simple square shape with edges that are 1 cm long. The area of this square is 1 cm times 1 cm, so 1 cm squared. But what happens if the shape is enlarged? If we enlarge the shape, as we did in the first example, by a scale factor of 2, each of our edges will now be 2 cm, it has doubled. However, what is the area of the square now? Be careful here, the area does not double as you might think. Instead, it will be 2 cm times 2 cm, which is 4 cm squared. Although the lengths were multiplied by 2, the area has been multiplied by a scale factor of 4. The rule of thumb is that when the lengths of a shape are multiplied by a constant scale factor, represented by k, then the area will be multiplied by k squared. Let's test it out. If a rectangle, which is 2 cm by 5 cm, is enlarged by 5, what will the area of the enlarged shape be? Is the answer a, 50 cm squared, b, 100 cm squared, or c, 250 cm squared? This is because the sides of our rectangle were increased by a scale factor of 5, making the new dimensions 10 and 25, and 10 times 25 gives you 250 cm squared. The easier way to calculate this is to work out the area of the original rectangle, so 2 times 5, which equals 10, and then multiply this by the square of the linear scale factor, so k squared equals 5 squared, which equals 25. You then multiply the two together to get 250 cm squared. We can also add another dimension. Imagine that instead of a 2D square, we have a cube, with each edge measuring 1 cm. The volume of this shape is 1 cm cubed. Let's say we enlarge each edge with a scale factor of 2. What will the new volume be? a, 3 cm cubed, b, 8 cm cubed, or c, 4 cm cubed. The correct answer is b. If we enlarge each edge of our cube by a scale factor of 2, our new volume will be 2 cm times 2 cm times 2 cm, or 8 cm cubed. The rule of thumb here is that when the lengths of a shape are multiplied by a constant scale factor represented by k, then the volume will be multiplied by k cubed. Let's try it with our rectangle example. Let's add another dimension to our 2 by 5 rectangle. Say the height is 10. We enlarge the shape by a scale factor of 5. What's the volume? We could work this out by calculating the enlarged lengths of each side and then multiplying them together. So 2 times 5 times 5 times 5 times 10 times 5, which equals 12,500 cm cubed. Or we can simply multiply the original volume. So 2 times 5 times 10, which equals 100 cm cubed, by k cubed, which is 5 cubed, which equals 12,500 cm cubed. Scale factors also apply to converting units. The metric system is a great example of this and is likely something you will be familiar with. For example, 1 liter can be converted into milliliters by multiplying the volume by a scale factor of 1,000. Another example is converting kilometres to millimetres. Say we have a distance that is 1 kilometre long. We would need to multiply 1 by 1,000, as there are 1,000 metres in a kilometre. Multiply that by 100, as there are 100 cm in a metre, and finally by 10, as there are 10 millimetres in a centimetre. That gives us a grand total of 1 million millimetres. It's handy to remember the following. If you liked the video, give it a thumbs up. And don't forget to subscribe. Comment below if you have any questions. Why not check out our Fuse School app as well? Until next time.