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Example: scaling of the area of a triangle
Question: In fig. (a), the larger triangle has sides twice as long.
How many times greater is its area.
Correct solution #1: Area scales in proportion to the square of the
linear dimensions, so the larger triangle has four times more area
(2
2
=4).
Correct solution #2: You could cut the larger triangle into four of
the smaller size, as shown in fig. (b), so its area is four times greater.
(This solution is correct, but it would not work for a shape like a
circle, which can’t be cut up into smaller circles.)
Correct solution #3: The area of a triangle is given by
A =
1
2
bh, where b is the base and h is the height. The areas of the
triangles are
A
1
=
1
2
b
1
h
1
A
2
=
1
2
b
2
h
2
=
1
2
(2b
1
)(2h
1
)
= 2b
1
h
1
A
2
/A
1
= (2b
1
h
1
)/(
1
2
b
1
h
1
)
= 4
(Although this solution is correct, it is a lot more work than
solution #1, and it can only be used in this case because a triangle is
a simple geometric shape, and we happen to know a formula for its
area.)
Correct solution #4: The area of a triangle is A =
1
2
bh. The
comparison of the areas will come out the same as long as the ratios
of the linear sizes of the triangles is as specified, so let’s just say
b
1
=1.00 m and b
2
=2.00 m. The heights are then also h
1
=1.00 m
and h
2
=2.00 m, giving areas A
1
=0.50 m
2
and A
2
=2.00 m
2
, so A
2
/
A
1
=4.00.
(The solution is correct, but it wouldn’t work with a shape for
whose area we don’t have a formula. Also, the numerical calculation
might make the answer of 4.00 appear inexact, whereas solution #1
makes it clear that it is exactly 4.)
Incorrect solution: The area of a triangle is A =
1
2
bh, and if you
plug in b=2.00 m and h=2.00 m, you get A=2.00 m
2
, so the bigger
triangle has 2.00 times more area. (This solution is incorrect
because no comparison has been made with the smaller triangle.)
The big triangle has four times more
area than the little one.
(a)
(b)
Chapter 1Scaling and Order-of-Magnitude Estimates