When we talk about scale factors, we are exploring something interesting in geometry. Scale factors can change how big or small shapes are and how much space they take up.

A scale factor is just a number that tells us how to make a shape larger or smaller.

For example, let’s look at a rectangle that is 4 cm long and 3 cm wide. To find its area, we do:

$$\text{Area} = \text{length} \times \text{width} = 4 \text{ cm} \times 3 \text{ cm} = 12 \text{ cm}^2.$$

Now, what if we use a scale factor of 2? This means we need to make the rectangle twice as big. We do this by multiplying both the length and the width by 2:

• New Length: $4 \text{ cm} \times 2 = 8 \text{ cm}$
• New Width: $3 \text{ cm} \times 2 = 6 \text{ cm}$

Let’s find the area of the new rectangle:

$$\text{New Area} = 8 \text{ cm} \times 6 \text{ cm} = 48 \text{ cm}^2.$$

Here’s something interesting: the new area is not just double the original area. It is actually four times bigger! This leads to a key rule:

When you change the scale factor, the area changes by the scale factor squared.

Quick Reminder: Area and Scale Factors

1. If the scale factor is ( k ), then:
   • The new area will be ( k^2 ) times the original area.

Using our example, since the scale factor was 2, we can confirm the new area of 48 cm(^2) like this:

$$\text{Original Area} \times 2^2 = 12 \text{ cm}^2 \times 4 = 48 \text{ cm}^2.$$

Example with a Scale Factor Less Than 1

Now let’s see what happens when we use a scale factor less than 1. Let’s take the same rectangle and use a scale factor of 0.5:

• New Length: $4 \text{ cm} \times 0.5 = 2 \text{ cm}$
• New Width: $3 \text{ cm} \times 0.5 = 1.5 \text{ cm}$

Now we find the area of the smaller rectangle:

$$\text{New Area} = 2 \text{ cm} \times 1.5 \text{ cm} = 3 \text{ cm}^2.$$

According to our area rule, when using a scale factor of 0.5, the area changes to:

$$\text{Original Area} \times (0.5)^2 = 12 \text{ cm}^2 \times 0.25 = 3 \text{ cm}^2.$$

In Summary

It’s important to understand how changing a scale factor affects the area of shapes. Here’s what to keep in mind:

• Scale Factor ( k ) means that area changes by ( k^2 ).

This idea works for all two-dimensional shapes and is very helpful for making scale drawings in your geometry studies!