Scaling of the area of a triangle

l / Example 1. The big triangle has four times more area than the little one.

In figure l, 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:

m / A tricky way of solving example 1, explained in solution #2.

You could cut the larger triangle into four of the smaller size, as shown in fig. (m), 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

(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 = 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 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 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 = bh, and if you plug in b = 2.00 m and h = 2.00 m, you get A = 2.00m², so the bigger triangle has 2.00 times more area. (This solution is incorrect because no comparison has been made with the smaller triangle.)