Now that you have investigated one type of polygon (rectangles), let's see if the relationship between the scale factor of a dilated polygon and the area holds true for other polygons and circles as well.
Recall that the perimeter of a circle is called the circumference.
Use the interactive sketch below to dilate Triangle A using the ScaleFactor slider. Notice how the dilated area changes as you manipulate the figures in the interactive.
Area of Triangle ABC |
Scale Factor |
Area of Triangle A'B'C' |
(Scale Factor)2 |
Area of Triangle ABC × (Scale Factor)2 |
2.4 |
3 |
21.61 |
9 |
21.6 |
2.4 |
2 |
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2.4 |
1.5 |
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|
2.4 |
0.75 |
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2.4 |
0.5 |
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Interactive popup. Assistance may be required.
What is the relationship between the area of triangle ABC, the scale factor, and the area of triangle A'B'C'?
Interactive popup. Assistance may be required. The area of triangle A'B'C' is equal to the product of the area of triangle ABC and the square of the scale factor.
Use the interactive sketch below to dilate Circle A using the ScaleFactor slider. Notice how the dilated area changes as you manipulate the figures in the interactive. You may also adjust the radius of Circle A by clicking and dragging Point B until the radius value changes to the desired number. Use this interactive sketch to complete the table beneath the sketch.
Area of Original Circle |
Scale Factor |
Area of Dilated Circle |
(Scale Factor)2 |
Area of Original Circle × (Scale Factor)2 |
12.57 |
2 |
50.27 |
4 |
50.28 |
12.57 |
1.75 |
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12.57 |
1.5 |
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|
12.57 |
0.75 |
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12.57 |
0.5 |
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Interactive popup. Assistance may be required.
What is the relationship between the area of the original circle, the scale factor, and the area of the dilated circle?
Interactive popup. Assistance may be required. The area of the dilated circle is equal to the product of the area of the original circle and the square of the scale factor.