Welcome to Omni's triangle inequality theorem calculator, where we'll answer the question "What is the triangle inequality theorem?" shortly at first and elaborately at second. In essence, the property describes the inequalities in one triangle, i.e., the triangle side length rules. Although fairly simple in itself, it has loads of important generalizations starting from the triangle inequality with absolute values and ending with the complicated Minkowski inequality and the even more complex Hölder inequality.

So, which are the possible side length of a triangle? Well, why don't we jump right into the article and find out?

Sides of a triangle rule: the triangle inequality theorem

Triangles are the simplest polygons in geometry: three sides, three inside angles, and that's all there is to them. Nevertheless, it proves enough to describe them accurately, e.g., by analyzing the trigonometric functions of the angles.

In essence, once you have a triangle at hand, there are several tools you can use to scrutinize it. However, we would like to look at the step before, i.e., at constructing the triangle. We all know that it's enough to have three line segments to build one, but can we always do it? After all, if we take two one-inch lines and a mile-long one, then it seems impossible. So which are the possible side lengths of a triangle?

The triangle inequality theorem states precisely when three line segments make up a triangle. The example above seems extreme, but it touches upon the problem: one side cannot be too long. Mathematically speaking, this means that certain inequalities in one triangle must be satisfied. Below, you have the theorem describing these triangle side length rules in symbols.

💡 If a, b, and c are three line segments, then they make up a triangle if and only if all the following inequalities are satisfied:
a + b > c, a + c > b, and b + c > a.

In fact, we can simplify the above sides of a triangle rule slightly.

💡 Three line segments make up a triangle if and only if the sum of the two shortest ones is larger than the third one.

This way, instead of checking three inequalities, we can limit ourselves to one - all to save some time since time is money. We do, however, have to be sure which segment is the largest.

The theorem doesn't seem too bad, does it? Still, mathematicians managed to push it further and came up with many generalizations: the triangle inequality with absolute value, the reverse triangle inequality, the Minkowski inequality, and the Hölder inequality. Essentially, they answer the question of "What is the triangle inequality theorem when there are no triangles around?".

We'll take the opportunity to talk about them all in later sections, but how about we see an example of the basic one first?

Example: using the triangle inequality theorem calculator

Say that you have a pile of planks lying around after the recent remodeling of your fence. It'd be a pity just to throw them away, so instead, you decide to put them to good use.

The idea is to construct a small storage cupboard for your tools. You'd like it to be triangular and fit in the corner of the garage. The planks you have come in four different lengths: 3 ft, 4.5 ft, 6 ft, and 9 ft. The problem is that you're not partial to sawing, so you'd prefer to keep the wood as it is.

So which are the possible side lengths of a triangle cupboard? It seems like it's time to shine for our triangle inequality theorem calculator and check which of the planks work!

Firstly, you check if the 3-, 6-, and 9-foot-long ones would do since that would even fit the rake for the blissful months when there are no leaves to take care of.

When we look at the triangle inequality theorem calculator, it has a picture of a triangle with sides a, b, and c, and three variable fields with those letters. Taking into account the plank lengths, we input:

a = 3 ft, b = 6 ft, c = 9 ft.

Once we give the last one, the triangle inequality theorem calculator spits out the answer underneath. Apparently, those three won't do, and indeed, the tool even tells us why: we have a + b ≤ c. In our case, it's 3 + 6 = 9 ≤ 9. (Note that we could have ordered the values differently. The basic answer would then be the same, but the inequality behind it would change.)

Let's have another go: take the 4.5-, 6-, and 9-foot-long ones. We change the values we input into the triangle inequality theorem calculator to have

a = 4.5 ft, b = 6 ft, c = 9 ft.

This time, the triangle side length rules are satisfied: the three lengths seem perfect for a cupboard!

But... is it going to fit nicely into the corner? For that, it would have to be a right triangle, wouldn't it? Oh, how lucky we are that Omni has a calculator to check that as well! Fingers crossed for no sawing!

Triangle inequality with absolute value and the reverse triangle inequality

Truth be told, whenever a high-brow mathematician hears the phrase "triangle inequality", it's not the theorem from the first section that comes to their mind first. Instead, it's the triangle inequality with absolute value.

💡 For any two numbers a and b, we have |a + b| ≤ |a| + |b|.

Let's point out a few crucial facts about this property.

  1. If a and b are non-negative, we get equality in the triangle inequality with absolute value.
  2. As a matter of fact, the property is much more general. Instead of numbers, a and b can be elements of a larger, multi-dimensional space, i.e., vectors. The symbol |a| would then mean the length of such a vector.
  3. In fact, we can go even further than that. The triangle inequality holds for any normed space. The elements of such a space can be, e.g., sequences or functions.

The triangle inequality is extremely useful in all fields of mathematics. To an extent, it describes spaces and sets that are "nice" in some sense. Also, it is one of the primary tools when proving various theorems. An example of such is the reverse triangle inequality.

💡 For any two numbers a and b, we have ||a| - |b|| ≤ |a - b|.

For our good ol' triangles, this translates to saying that a triangle's side is always greater than the difference between the other two. However, in general, the reverse triangle identity (just like the non-reversed version) works in any normed space.

We hope that this section sparked your interest in higher mathematics. And if so, then let us travel even higher in the next section.

Minkowski inequality and Hölder inequality

Seeing the sides of a triangle rule from the first section and the generalizations from the above section, ** a curious mind could ask**, "What if we have more numbers? What is the triangle inequality theorem then?" Well, this section gives the answer.

Let's start with the Minkowski inequality.

💡 For p ≥ 1 and elements a₁, a₂,..., aₙ and b₁, b₂,..., bₙ, we have
(|a₁ + b₁|p + ... + |aₙ + bₙ|p)1/p ≤ (|a₁|p + ... + |aₙ|p)1/p + (|b₁|p + ... + |bₙ|p)1/p.

Note that the fractional exponent of 1/p is, in fact, the p-th root. Also, observe that for p = 1 and n = 1, we obtain precisely the triangle inequality with absolute value from the above section.

And last but not least, let's see the most complex theorem of this article: the Hölder inequality.

💡 For p,q ≥ 1 satisfying 1/p + 1/q = 1 and elements a₁, a₂,..., aₙ and b₁, b₂,..., bₙ, we have
|a₁ · b₁| + ... + |aₙ · bₙ| ≤ (|a₁|p + ... + |aₙ|p)1/p · (|b₁|q + ... + |bₙ|q)1/q.

Note that this one, in comparison with the others, deals with multiplication rather than addition. Also, observe how on the left side, we don't have any p or q the way we did in the Minkowski inequality. That means that when we use the Hölder inequality, we can always adjust them to best suit our purpose.

Phew, we've come a long way, haven't we? From inequalities in one triangle to complicated theorems from high school mathematics. If you've read through the whole article, you most certainly deserve a short nap and a treat. We don't know about you, but for us, some hot chocolate would do just fine.

Maciej Kowalski, PhD candidate