PERIMETERS and AREAS of SIMILAR POLYGONS

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PERIMETERS and AREAS of SIMILAR POLYGONS

Suppose that a triangle with sides a , b , and c has been scaled by s
to get a similar triangle with corresponding sides A , B , and C .
Thus, A=sa and B=sb and C=sc .

Let's investigate the relationship between the perimeters of these two triangles:

perimeter of original triangle= a+b+c

perimeter of scaled triangle
    =A+B+C
    =sa+sb+sc
    =s(a+b+c)
    =s(perimeter of original triangle)

Thus, the perimeter ends up being scaled by the same factor that scales the sides!
A similar calculation shows that this result is indeed true for polygons in general:

THEOREM: Perimeters of Similar Polygons
The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides.

Area, on the other hand, behaves a bit differently.
Suppose you have a square of side a that has been scaled by s to get a square of side A .
Thus, A=sa .

Let's investigate the relationship between the areas of these two squares:

area of original square= a⋅a=a2

area of scaled square
    =A⋅A
    =(sa)⋅ (sa)
    =s2⋅a2
    =s2(area of original square)

Thus, the area ends up being scaled by the square of the factor that scales the sides!
This idea leads to the following theorem:

THEOREM: Areas of Similar Polygons
The ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides.

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