Greenwood press

SIMILARITY
121
Dimensions of a standard television
screen with a 4-to-3 aspect ratio. 

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Similarity can be used to approximate lengths and distances. For example, on
a sunny day you can use similarity to determine the height of a tall object such
as a flagpole by using just a tape measure. If you measure your height, your dis-
tance from the flagpole, and the length of your shadow, then you will be able to
set up a proportion to find the height of the flagpole. For instance, suppose you
are standing 5 meters away from the flagpole, you are 1.65 meters tall, and you
measure your shadow to be 1.38 meters long (see the figure below). Similar tri-
angles can be used to show that your height corresponds to the flagpole height,
and your shadow length corresponds to the flagpole’s shadow length.
In this case, the proportion 
f
5+1.38
=
1.64
1.38
can be used to find the height of the
flagpole,
f , which equals approximately 7.63 meters (close to 25 feet).
Architects and designers use similarity to create and visualize new buildings.
A miniature two- or three-dimensional model that is a replica of a future building
is often put together during a design phase. It is easier and less expensive to make
changes to a miniature replica of an object than to the object itself, so careful
attention to size and detail is important in model-making. Once the ideas behind
the design of the house are negotiated, the floor plans are passed on to the builders
to replicate the model on a larger scale. Since the actual floor space of the house
is similar to the paper mock-up of the floor plan, the corresponding dimensions
between the real structure and the model are proportional. However, the area com-
paring the house’s floor space to the floor-plan area is proportional to the square
of the ratio of the dimensions. For example, if the house is 50 times larger than
the floor plan, then the area of the house is 2,500 (which is 50
2
) larger than the
floor plan. This area proportion of similar figures is squared, because area is a
measurement of two dimensions. For example, suppose two similar squares have
respective lengths of 2 and 100 cm. The area of the squares would be 4 cm
2
and
10,000 cm
2
, respectively. Even though the ratio of their lengths is 100/2 or 50, the
ratio of their areas is 10,000/4 or 2,500, which is the same as 50
2
. Carpenters can
use this information to determine the amount of wood and carpeting needed for
the floors if they are not given the actual dimensions of the house.
Similarity can also be used to predict the mass of unusually large or even
extinct animals, such as dinosaurs. A scale model of a dinosaur can be used to
predict the actual volume of it, assuming that the ratio comparing the actual
length to the model length is available. Suppose that an accurately scaled model
of a tyrannosaurus with a length of 0.3 meters is used to determine its mass.