topic 3 scale factors and areas of 2-d shapes unit 8 topic 3
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Topic 3Scale Factors and Areas of 2-D Shapes
Unit 8 Topic 3
ExploreAny shape can be enlarged or reduced by multiplying each of its dimensions by the same linear scale factor. 1. Work with a partner to complete the table
below.
2. What is the relationship between the linear
scale factor and the area scale factor?
Try this on your own first!!!!
Explore1.
2. What is the relationship between the linear
scale factor and the area scale factor?
Try this on your own first!!!!
You should notice that the area scale factor is equal to the linear scale factor squared.
4 9 16 25 36
InformationThe relationship between the area of a new shape and the area the original shape can be expressed using the following equation.Area Scale Equation
new area = old area • k2
where k is the linear scale.
We can rearrange the equation to isolate the area scale factor, k2. Area Scale Factor (ASF)
2 new areaASF = =
old areak
Example 1Determining a New Area
Maggie scanned an 8” by 10” photograph of a hummingbird to her computer so that she could change the size. a) If the photograph is enlarged by a linear scale
factor of 4, then determine the area of the enlarged photograph.
Method 1: Using Area CalculationFind the area of the enlarged picture.
Try this on your own first!!!!
Example 1a: SolutionDetermining a New Area
Method 1: Using Area CalculationFind the area of the enlarged picture.
new length = old length
new length = 8 in. 4
new length = 32 in.
k
new width = old width
new width = 10 in. 4
new width = 40 in.
k
2
32 40
1280 in .
A l w
A
A
Example 1a: SolutionDetermining a New Area
Method 2: Using the Area Scale EquationSubstitute into the area scale equation
2area of new object = area of old object k
2
8 10
80 in .
old
old
old
A l w
A
A
2
2
2
area of new object = area of old object
area of new object = 80 4
1280 in .new
k
A
Example 1b: SolutionDetermining a New Area
b) Suppose Maggie decided to decrease the size of the original photograph by a linear scale factor of . What is the area of the reduced image?
12
2
2
2
area of new object = area of old object
1area of new object = 80
2
20 in .new
k
A
Example 1c: SolutionDetermining the Area Scale Factor
c) Determine the area scale factor if the linear scale factor is . 1
2
2
2
2
ASF = LSF
ASF =
1ASF =
2
1ASF = or 0.25
4
k
Example 2Determining the scale factor of an enlargement
Try this on your own first!!!!
Jim’s laptop has a monitor with the dimensions 9 in by 12 in. The image on his laptop is projected onto a screen. The image on the screen, which is similar to that on the laptop, has an area of 2 700 in2. By what factor did the area of the screen increase by? (That is, how many times greater is the area of the screen than the laptop?)
Example 2a: SolutionJim’s laptop has a monitor with the dimensions 9 in by 12 in. The image on the screen, which is similar to that on the laptop, has an area of 2 700 in2. By what factor did the area of the screen increase by?
2
2
2
2
area of new object = area of old object
2700 108
2700 =
10825= = ASF
k
k
k
k
2
9 12
108 in .
old
old
old
A l w
A
A 22700 in .newA
Example 2b: SolutionDetermining the scale factor of an enlargement
Determine the linear scale factor used to project the image from the laptop to the screen.
2
2
25= = ASF LSF=
25
5 LSF
k k
k
k
Example 3Determining the Area Given a Scale Diagram
Mr. and Mrs. Smith recently moved into a new home. In their rectangular backyard, they have a rectangular patio and a circular koi fish pond in the corner, as shown in the scale diagram below.
a) If the radius of the pond in the diagram is 1.5 cm, what is the area of the diagram koi pond, to the nearest tenth?
Try this on your own first!!!!
Example 3a: SolutionDetermining the Area Given a Scale Diagram
Mr. and Mrs. Smith recently moved into a new home. In their rectangular backyard, they have a rectangular patio and a circular koi fish pond in the corner, as shown in the scale diagram below.
a) If the radius of the pond in the diagram is 1.5 cm, what is the area of the diagram koi pond, to the nearest tenth?
2
2
2
1.5
7.1 cm
new
new
new
A r
A
A
Example 3b: SolutionDetermining the Area Given a Scale Diagram
Determine the area, to the nearest tenth of a square cm, of the actual pond if the diagram was drawn using a linear scale factor of 0.01.
2
2
2
area of new object = area of old object
7.1 = area of old object 0.01
7.1 = area of old object 0.0001
7.1 = area of old object
0.000171000cm = area of old object
k
Need to Know:• The area scale factor, ASF, of a 2D shape is
• The area of the original or old shape is multiplied by the area scale factor to produce the area of the new shape.
• The area scale equation is new area = old area • k2,
• where k is the linear scale factor.
You’re ready! Try the homework from this section.
2 new areaASF = =
old areak