1. Area and Perimeter of Similar Figures
10.4
Learning Target
I can…...
… find the perimeter and area of an object given the perimeter and area of a similar object and the dilation factor.

2. Effect of Dilation on Perimeter
The perimeter of the blue rectangle to the left below is 10 ft ( )
2 ft
Dilate the figure by a factor of 3
3 ft
6 ft
9 ft
The dimensions are tripled, resulting in a new perimeter. The perimeter of the scaled rectangle is or So, if we dilate an image by any factor, the perimeter is changed by the same dilation factor or ratio

3. Effect of Dilation on Perimeter
Example 1: A geometric figure with a perimeter of 12 meters is scaled(dilated) by a factor of 2.5.
Result: The perimeter of the resulting image is 12*2.5 or 30 meters
Example 2: The same original object is dilated by a factor of 2/3.
Result: The perimeter of this resulting image is 12 *(2/3) or 8 meters

4. Effect of Dilation on Area
The area of the blue rectangle to the left below is 6 sq. ft (3  2)
2 ft
Dilate the figure by a factor of 3
3 ft
6 ft
9 ft
The dimensions are tripled, resulting in a new area. The area of the scaled rectangle is 9  6 or 54 sq. ft. So, if we dilate an image by a factor of 3, the area is changed by a factor of 9 or 32

5. Effect of Dilation on Area
To see this more clearly, let’s see how many blue rectangles fit within the red/scaled rectangle. 1 2 3 4 5 6 7 8 9
The area of the scaled (red) rectangle is 9 times that of the blue because 9 of the blue rectangles are needed to fill the same space as the red rectangle which was the blue rectangle dilated by a factor of 3.

6. Effect of Dilation on Area
Example 1: A geometric figure with an area of 24 sq. inches is scaled(dilated) by a factor of 2.
Result: The area of the resulting image is (24)  (22) 24  4 = 96 sq. inches
Example 2: The same original object is dilated by a factor of 1/2.
Result: The area of this resulting image is 24  (1/2)2 24  (1/4) = 6 sq. inches

7. Prep for Homework
The figures to the left are similar. What are the ratios of the perimeter and the area? 3 5
The ratio of the side lengths is 5:3 or 5/3
The ratio between perimeters is also 5:3 or 5/3
The ratio between areas is therefore (5:3)2 or (5/3)2
The result is a ratio of 52:32 or 52/32  25:9 or 25/9

8. Prep for Homework
The figures to the left are similar. The smaller figure has an area of 3 sq. ft. and the other has an area of 48 sq. ft. What is the scale(dilation) factor between the objects?
The ratio of the areas is 48:3 or 48/3 which equals 16.
Since the ratio of the areas is the square of dilation factor, the dilation factor is the square root of the area ratio.
The scale factor is therefore the square root of 16 or 4. That means the ratio between perimeters is also 4.