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If the ratio of the areas of two squares is 9:16, what is the ratio of their perimeters?
3:4
9:16
4:3
3тАЛ:2
4
Since the ratio of areas of similar figures is the square of the ratio of their corresponding linear dimensions (like side or perimeter), taking the square root of the area ratio directly gives the perimeter ratio in 5 seconds.
Ratio of the areas of two squares = 9:16
Area2тАЛArea1тАЛтАЛ=(s2тАЛs1тАЛтАЛ)2=(P2тАЛP1тАЛтАЛ)2
Since the ratio of areas of similar figures is the square of the ratio of their corresponding linear dimensions (like side or perimeter), taking the square root of the area ratio directly gives the perimeter ratio in 5 seconds.
Students often confuse the area ratio with the perimeter ratio and mistakenly choose option B (9:16), forgetting that area scales quadratically while perimeters scale linearly.
Recall the Area and Perimeter Relationship
For any two similar figures or squares, the ratio of their areas is equal to the square of the ratio of their sides or perimeters.
Area2тАЛArea1тАЛтАЛ=(P2тАЛP1тАЛтАЛ)2
Substitute the Given Values
Substitute the given area ratio of 9:16 into the equation to set up the ratio of perimeters.
169тАЛ=(P2тАЛP1тАЛтАЛ)2
Calculate the Square Root
Take the square root on both sides to find the ratio of their perimeters.
P2тАЛP1тАЛтАЛ=169тАЛтАЛ=43тАЛ
A is correct because the ratio of the perimeters is the square root of the ratio of the areas, which equals 43тАЛ.
This scaling principle applies to all similar 2D shapes (circles, triangles, polygons) where the ratio of areas is the square of the scale factor, and 3D volumes scale with the cube of the scale factor.
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