k / Example 1.
The big trian-
gle has four times more area than
the little one.
l / A tricky way of solving ex-
ample 1, explained in solution #2.
One of the first things I learned as a teacher was that students
were not very original about their mistakes. Every group of students
tends to come up with the same goofs as the previous class. The
following are some examples of correct and incorrect reasoning about
Scaling of the area of a triangle
. In figure k, 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
(22 = 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 = bh/2, where b is the base and h is the height. The areas of
the triangles are
A1 = b1h1/2
A2 = b2h2/2
A2/A1 = (2b1h1)/(b1h1/2)
(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
Correct solution #4: The area of a triangle is A = bh/2. 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 b1 = 1.00 m and b2 = 2.00 m. The heights are then also
h1 = 1.00 m and h2 = 2.00 m, giving areas A1 = 0.50 m2 and
A2 = 2.00 m2, so A2/A1 = 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 cal-
culation 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 = bh/2, and i