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Here is a rectangle with a length of 5 units and a width of 2 units.
| scale factor | area of image in square units | factor by which the area changed |
|---|---|---|
| 0.5 | ||
| 1 | ||
| 2 | ||
| 2.5 | ||
| 3 | ||
| 4 |
Andre says, “We know that if a rectangle is scaled by a factor of , the area scales by a factor of . Does this apply to other shapes?”
Jada says, “Here’s a shape that’s not a rectangle. Say its area is square units. Let’s draw some rectangles on it that get smaller and smaller to fit the remaining empty space. With enough rectangles we can come close to covering the whole blob.”
Andre says, “These rectangles start to make a nice approximation of the blob. If we wanted to get closer, we could add even more rectangles. The sum of the areas of all the rectangles would add up to the area of the blob. I think we’re almost there!”
Here is a 5-unit-by-4-unit rectangle. Its area is 20 square units. What will the area be after it is dilated by a scale factor of 3?
The dilated rectangle’s dimensions can be written as and . Substitute these values into the area expression to get . Rearrange the numbers to get . This can be rewritten or . So the area of the dilated rectangle is 180 square inches, or 9 times the original. The diagram confirms that 9 of the original rectangles fit into the dilated rectangle.
In general, when scaling a rectangle by a factor of , the length and the width are both multiplied by , so the area is multiplied by .
What about a shape that is not a rectangle? Consider the rounded shape called an ellipse in the image. We can approximate the area of an ellipse by filling it with many rectangles. The sum of the areas of the rectangles will be a little less than the area of the ellipse because they don’t fill it entirely. To get closer to the area of the ellipse, draw in more rectangles. Continuing the process infinitely would give the exact area of the ellipse.
When this ellipse is dilated using a scale factor of 4, each rectangle becomes 16 times larger. This means that the area of the ellipse increases by a factor of 16 as well. Any closed shape can be filled with rectangles that approximate its area. Because of this, when any shape is scaled by a factor of , its area is multiplied by a factor of .