Triangle Inequality Theorem

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The Triangle inequality theorem is one of the major mathematical concepts that outlines how the triangle works. The theorem is very important for algebraic and real-life concepts. Surveyors use the theorem for urban planning and transportation as it can help them get a rough dimension of the magnitudes of certain land space. However, the scope of this piece is limited to the theorem and its relationship to the triangle. This is so that students can know exactly what they need to do anytime they have to define a triangle.

Generally, the understanding of a triangle is any polygon that has three enclosed sides. While this is not actually wrong, the fact is that it is only a part of what makes a triangle one. In fact, there are more important rules that any three sides enclosed polygons must fulfil for them to be considered a triangle. The triangle inequality theorem reveals and highlights these conditions to help students know more about the triangle.

What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that for any three-sided enclosed polygon to be considered a real Triangle, the sum of the length of any two sides must be greater than the last side.

This theorem means that irrespective of the length of a triangle, no length should be big enough such that it is greater than the sum of the length of the other two sides. The good thing about this theorem is that there is no real way a single length of a side can be greater than the length of two sides.

The Properties of Triangle Inequality

  • The sum of any two sides of a triangle is greater than the last side
  • If the addition of any two sides of a triangle is greater than the last side, then by implication, the differences of both sides must be lesser than the last side
  • In a triangle, the length directly opposite the largest angle is generally the longest among all three lengths.

Example 1

Now, let us consider a triangle ABC

Based on the Triangle Inequality Theorem

  • Length AB + length BC will be greater than AC
  • Length AB + length AC will be greater than BC
  • Length AC + length BC will be greater than AB

If any of the three conditions are not true, then the above-enclosed polygon cannot be considered a triangle.

Example 2

We will consider a second example to show why the two lengths sum of a triangle should be smaller than the two sides. Let us use the triangle below as an example.

Now if we consider all the addition of the sides we will see that

Length AB + length BC will be greater than AC

6 + 9 = 15. 15 > 13

Length AB + length AC will be greater than BC

6 + 13 = 19. 19 > 9

Length AC + length BC will be greater than AB

13 + 9 = 22. 22 > 6

Now let us imagine that BC, which is equal to 9 cm, was actually 5 cm. That will give us a triangle that looks something like this

Clearly 6 + 5 = 11 and 11 < 13.

From the triangle above, it is clear that there is not possible that we can create an intersection between BC. Also, even if we graphically join AB and BC together, they won’t still be long enough to become to meet AC. 11 CM will be just short of 13 cm. so in essence, it is impossible to form a triangle where two side lengths are lesser than one long side.

Just like we have shown above, making AC less than the differences between AB and AC will make it impossible to form a triangle as well. Students can attempt to make a measurement to prove this.

Example 3

Now let us consider a triangle where the length of two sides is 13 cm and 18 cm, respectively. What is the possible long-range the third side will fall into?


We will need to consider the properties of the triangle inequality theorem to get the right answer. Recall that the second property stated that;

If the addition of any two sides of a triangle is greater than the last side, then by implication, the differences of both sides must be lesser than the last side

Now 13 cm + 18 cm = 31 cm


18 cm – 13 cm = 5 cm

Since the differences and sum of the two known lengths are 5 cm and 13 cm, then the third length can be from 13.1 cm to 30.9 cm

Proof of the Triangle Inequality Theorem

The Triangle Inequality Theorem can be proven mathematically. While some of the computations above have shown how the inequality theorem works, students can know the real proof.

We will consider a triangle given as ABC

Our aim is to show that AB + AC is greater than BC

For that to be possible, we will have to graphically add the same length of AC to AB and term it AD.

The new addition will leave us with a triangle ABD AND Triangle ADC where BD is actually AB + BC.

Now considering the graphical visualization of our above triangle, it is obvious that BD is indeed larger than BC. Also, if the angle of triangle ABC is known, it is obvious that it will be smaller than the triangle BDC. The larger the angle, the larger the sides opposite it. This theory clearly favors length BD as a bigger side compared to length BC.

This simple proof can be used to test the other two sides against another, and the answer will remain consistent, justifying the Triangle Inequality Theorem.

Triangle Inequality Theorem Calculator

The triangle inequality theorem is easy to understand when considered from all angles. As shown in the above examples, the inequality theorem emphasizes the conditions on which the triangle can be considered as such. Generally, the calculations based on the theorem are easy to compute and do not require much.

However, it is possible for students to use mathematical software and calculators to get some of the solutions. There is more than one online triangle inequality theorem calculator on the internet, and it saves time if students use them when necessary.


The Triangle Inequality Theorem is an important concept that many students pursuing mathematics as a degree or going into geography will have to deal with. This piece has considered the major aspects that students will have to know when it comes to this theorem. The three properties of the theorem were also stated.

These properties are what establish whether a polygon is indeed a triangle or not. The proof of the theorem was also highlighted to show how the triangle inequality theorem works. Several examples were also considered in this piece, and the good thing is that students can try their hands on several problems to prove whether a three-sided enclosed polygon is indeed a triangle. It is important to state that there are many online examples to try the Triangle Inequality theorem.

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