Unit 5, Lesson 4
Scaling and Area
Geometry
4.2
Activity
Scaling Up a Rectangle
Here is a rectangle with a length of 5 units and a width of 2 units.
- What is the area of the rectangle?
- Dilate rectangle
using center and a scale factor of 2. Calculate the area of the image. - Dilate rectangle
using center and a scale factor of 3. Calculate the area of the image. - Complete the table.
scale factor area of image in square units factor by which the area changed 0.5 1 2 2.5 3 4 - Write an expression for the area of a rectangle with length
and width . - Imagine dilating the rectangle with length
and width by a factor of . Write expressions for the dimensions of the dilated rectangle. - Write an expression for the area of the dilated rectangle.
- Use your work to draw a conclusion about what happens to the area of a rectangle when it’s dilated by a scale factor of
.
4.3
Activity
What about Other Shapes?
Andre says, “We know that if a rectangle is scaled by a factor of
Jada says, “Here’s a shape that’s not a rectangle. Say its area is
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!”
- Suppose the blob is dilated by a factor of
. In doing this, the rectangles covering the blob also get dilated by a factor of . How does the area of each dilated rectangle compare to the area of each original rectangle? - What does this tell you about the area of the dilated image? Explain your reasoning.
- Suppose a circle has area 20 square inches and it’s dilated using a scale factor of 6. What is the area of the image? Explain or show your reasoning.
Student Lesson Summary
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
In general, when scaling a rectangle by a factor of
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
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