Most of you understand that the scale factor is the scale length over the original length. But what happens if you're looking for the area and don't know the length? To find out, let's first use scale factor to find area of a new shape.
This rectangle has an area of 3 * 8 or 24. If we dilate it with a scale factor of 2, the new sides will be multiplied by two. But is the area also multiplied by 2? Let's find out if we get the same thing. 24 * 2 = 48. But 3 * 2 = 6 and 8 * 2 = 16. Multiplying 6 * 16 gives us 96! 96 does not equal 48, so something else must have happened. Let's compare 96, the scale area to 24, the original. If we divide 96 by 24 we get 4, not 2. So something happened to the scale factor when we calculated area. It got squared! 2 * 2 = 4.
This means that any time we are working with area only and we have a scale factor to find new area, we need to square it. Also, we know that area is two-dimensional and that units that are two dimensional are square. Therefore it makes sense to square it. It also leads to the next thought.
What do we do with scale factor and volume?
