A Unique Physical Derivation of Heron's Formula

Basic Assumptions

As mentioned, the thinking and methods of this derivation have a "physical flavor." What does it mean to have a "physical flavor"? In theoretical physics, physicists often derive a complete physical law based on some very basic assumptions and then test whether it matches experimental results. Einstein and his theory of relativity can be considered pioneers of this approach. Now, we apply a similar line of thought, based on some basic assumptions, to directly construct Heron’s formula.

Let the lengths of the three sides of the triangle be a, b, c, and the formula for the area of the triangle be S(a, b, c).

Basic Assumptions

1. Area has the dimension of length squared. This is perhaps the most fundamental principle.

2. S(a, b, c) is symmetric with respect to a, b, c. We have no reason to reject this, as we find no side to be more special than the others. Moreover, symmetry actually helps us discover the correct formula.

3. The form of S(a, b, c) should be as concise as possible. This is an imposed condition; physicists hope that physical laws are as concise as possible, so they include the principle of simplicity when deriving physical laws—that is, among all possible physical laws, the simplest one is chosen. Whether this principle holds in mathematics is open to debate, but in the case of Heron’s formula, it indeed holds.

Known Facts

The formula we derive must be consistent with some simple facts we already know; this is the requirement of compatibility. Some facts we know can be listed as follows:

4. When the sum of two side lengths equals the third side length, the area is 0, i.e., S(a, b, a+b) = 0; specifically, when any side is 0, the area is 0.

5. The area formula for a right-angled triangle is known to us, namely S(a, b, \sqrt{a^2+b^2}) = \frac{1}{2}ab.

Derivation Process

First, we start from symmetry. There are many symmetric expressions that can be constructed from the three sides, the simplest being abc. However, this clearly does not satisfy the first assumption, as its dimension is length cubed rather than length squared. A simple way to improve it is to consider: (abc)^{2/3} This formula is decent; it satisfies the three assumptions, and when any side is 0, the area is indeed 0. However, it cannot fully satisfy Fact 4, because when a=1, b=1, c=2, (abc)^{2/3} is clearly not 0, so this formula is excluded. Of course, there are many others one could think of, such as ac+bc+ab, which also satisfies the three assumptions, but when one side is 0, it does not result in an area of 0, so it is also excluded.

Fact 4 is a very strong condition. To satisfy Fact 4, we hope that when the sum of two sides equals the third side, a zero will appear. The simplest way to achieve this is a+b-c, and according to the requirement of symmetry, we have: (a+b-c)(b+c-a)(c+a-b) This expression satisfies symmetry and Fact 4, but it does not satisfy the basic dimensional requirement. We could take the cube root of its square, or multiply it by a length and then take the square root. As for how to proceed, we determine this based on Fact 5.

This brings us to the most crucial part of this article, and the most "physical" part. Fact 5 is a test regarding right-angled triangles. We could directly substitute c = \sqrt{a^2+b^2}, but because of the square root (a situation common in physics), it becomes cumbersome (or even impossible to proceed). Therefore, we do not consider the full case, but only the infinitesimal case, assuming b is an infinitesimal quantity, such that: c = \sqrt{a^2+b^2} \approx a + \frac{b^2}{2a} This differs from a only by a second-order infinitesimal. Thus, at first-order precision, c = a. The area of a right-angled triangle is \frac{1}{2}ab. In other words, for a triangle with side lengths a, b, a where b is infinitesimal, the area is \frac{1}{2}ab at first-order precision. Substituting a, b, c=a into (a+b-c)(b+c-a)(c+a-b), we get: b \times b \times (2a-b) \approx 2ab^2 We find that if we simply multiply by a and then take the square root, we can obtain ab. However, multiplying by a alone does not satisfy symmetry. The simplest way is to multiply by (a+b+c). Thus, we obtain a candidate formula: \sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)} The leading coefficient needs to be adjusted. If we directly substitute a, b, c=a, we get: \sqrt{(2a+b) \times b \times b \times (2a-b)} \approx \sqrt{4a^2b^2} = 2ab This differs from \frac{1}{2}ab by a factor of \frac{1}{4}. Therefore, a candidate formula that basically satisfies the three assumptions and two facts is: S(a, b, c) = \frac{1}{4}\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}

This is exactly the correct Heron’s formula! Up to this point, we have "vividly" constructed Heron’s formula! Of course, from the perspective of overall logical reasoning, this is only a candidate—a very likely formula—and its correctness still needs to be proven. Next would be the proof, which is a matter of rigor and does not involve advanced content, so it is not provided here.

Why Do This?

Why do this for a formula that is already widely known? This process is neither a rigorous proof nor does it yield anything new; what is its significance? For practical-minded friends, at least for now, this task has no significance. However, there may still be some unexpected benefits.

First, doing this—or rather, using different methods to accomplish the same thing—helps us understand the essence of the matter. By comparing various lines of thought, we discover the strengths and weaknesses of each, thereby finding the key points of every approach. Secondly, this is a simulated derivation process, or rather, a process of discovering a new formula. This kind of process is of great significance in guiding physical discovery. As an enthusiast of mathematics and physics, I naturally hope that this line of thinking can also bring some fresh vitality to mathematics—though, of course, this is merely an attempt.