Any shape is similar to another if all corresponding measurements are
in proportion to a constant, called a scaling factor. Similar figures
are such that by zooming in or zooming out of one figure, we obtain a
figure identical to the other figure. Similar figures have the same
shape.
Similar Triangles
Triangles are similar when their angles are the same. (We don't need to
say they are corresponding, because if two angles are the same, the
third one will also be the same, and the sides will be corresponding).
In the diagram below:
ABC is a triangle, and DE is parallel to AB. DEC is therefore similar to ABC. The height of DEC is h1,
and the height of ABC is h. Because they are similar triangles, their
corresponding dimensions are proportional to a scaling factor, so:
So:
For instance, if h=4, h1=2, AB=4 then h1=2/4*2=2, that is it is half the size. Or because h1/h=2/4, the scaling factor is 1/2.
However, the area of the big triangle is 4*4//2=8, and the area of the
smaller triangle is 2*2/2=2. That is, the ratio of the areas is 1:4.
In general, if s is the scaling factor, and DE=b1 and AB=b, then
If A1 is the area of DEC, and A is the area of ABC, then:
Now, because h1/h=b1/b=s, then
Any Similar Figures
The ratio of the areas of any similar figures are in proportion to the scaling factor squared.
Suppose that the two patterns are similar; that is, all their
corresponding dimensions are proportional to the scaling factor, s.
Suppose we represent the areas by small squares (pixels) of side x, so
the area of each square is x2, then the area, A1, covered by
one square in one of the similar figures will be covered by an area,
A2, sx*sx in the other figure. We can, of course, make our little
squares so small, they exactly represent the areas. The ratio of the
areas, A1 and A2 is therefore:
And any similar figures, or patterns are therefore proportional to the scaling factor squared.
