Two rectangles are similar. If the height of the first rectangle is 3 inches, and the height if the second rectangle is 9 inches, how much larger is the second rectangle's area?
areas of similar figures are proportional to the square of their sides
area1 : area2 = 3^2 : 9^2
= 9 : 81
= 1 : 9
To find the ratio of the areas of two similar rectangles, you can use the concept of scale factor. The scale factor is the ratio of any corresponding lengths of the two rectangles. In this case, we are given the heights of the two rectangles.
The scale factor can be determined by dividing the height of the second rectangle by the height of the first rectangle:
Scale factor = Height of second rectangle / Height of first rectangle
Scale factor = 9 inches / 3 inches = 3
Since the two rectangles are similar, all corresponding lengths are in the same ratio. This means that the scale factor can be applied to the length as well as the height. Therefore, the scale factor can also be used to compare the areas of the rectangles.
To find the difference in the areas, we need to square the scale factor.
Scale factor squared = 3^2 = 9
Therefore, the second rectangle's area is 9 times the area of the first rectangle.
In summary, the second rectangle's area is 9 times larger than the first rectangle's area.
To find the ratio of the heights of the two rectangles, we divide the height of the second rectangle by the height of the first rectangle:
Ratio of heights = height of second rectangle / height of first rectangle = 9 inches / 3 inches = 3
Since the rectangles are similar, the ratio of their areas is the square of the ratio of their heights.
Ratio of areas = (ratio of heights)² = 3² = 9
Therefore, the second rectangle's area is 9 times larger than the first rectangle's area.