Question 1004241: The areas of two similar polygons are in the ratio 36:16. Find the ratio of the corresponding sides?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the ratio of the sides will be equal to the square root of the ratio of the areas.
if the ratio of the areas is 36/16, then the ratio of the sides will be 6/4.
a couple of examples should confirm this.
assume a square.
s^2 = area.
let the large square have a side length of 6 and the small square have a side length of 4.
6^2 = 36
4^2 = 16
assume a rectangle.
let the length and width of the large rectangle be equal to 12 and 6 and the small rectangle have a length of 8 and 4.
area of large rectangle = 12 * 6 = 72
area of small rectangle = 8 * 4 = 32
ratio of area of large rectangle = 72/36 = 36/16.
the ratio of the sides is 6/4 and the ratio of the areas is 36/16.
6/4 is the square root of 36/16.
this applies to all polygons.
for example:
assume an octagon (8 sides).
the area of the octagon is equal to 2 * (1 + sqrt(2)) * a^2
a is the side of the regular octagon.
it's clear that if a is 6, then the area will be equal to 36 * the constants, and that if a = 4, the area will be equal to 16 * the constants.
take the ratio of the larger area to the smaller area and it will becomes 36/16 because the constants, being the same value, will cancel out.
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