Welcome to Calcugo's triangle inequality theorem calculator, where we'll answer the question "What is the triangle inequality theorem?" — shortly at first, and elaborately after. 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 lengths 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 words: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If even one of these three inequalities fails, the three segments cannot close up into a triangle. If a sum is exactly equal to the third side (e.g., a + b = c), the "triangle" collapses into a straight line — this is called a degenerate triangle.
Example: using the triangle inequality theorem calculator
Suppose someone hands you three sticks measuring a = 7, b = 10, and c = 5 units, and asks whether they form a triangle. Let's check all three inequalities:
- a + b > c → 7 + 10 = 17 > 5 ✓
- a + c > b → 7 + 5 = 12 > 10 ✓
- b + c > a → 10 + 5 = 15 > 7 ✓
All three conditions hold, so yes — these sides form a valid triangle. Because all three sides are different, it is a scalene triangle. The calculator confirms this instantly and, for a valid triangle, also reports its perimeter, semiperimeter, area (via Heron's formula), and its classification by angles (acute, right, or obtuse).
To use the calculator, just:
- Pick your unit system (metric or US/imperial) and the unit of measurement.
- Enter the three side lengths a, b, and c.
- Read the verdict, the check of all three inequalities, and — for the third side given the other two — the allowed range.
Triangle inequality with absolute value and the reverse triangle inequality
The triangle inequality also tells us, if we know two sides, what the possible length of the third one can be. Rearranging a + b > c and b + c > a for the same side gives the reverse triangle inequality, which bounds each side from both above and below:
|a − b| < c < a + b
In other words, the third side must be shorter than the sum of the other two, but longer than their absolute difference. The absolute value here guarantees the lower bound is never negative, no matter which of the two known sides is larger. The general form of the triangle inequality for real numbers or vectors is written as:
|x + y| ≤ |x| + |y|, and | |x| − |y| | ≤ |x − y|
Minkowski inequality and Hölder inequality
The humble triangle inequality is a special case of much deeper results in mathematical analysis. The Minkowski inequality generalizes it to the p-norm of sequences or functions. For p ≥ 1, and vectors x and y:
∥x + y∥p ≤ ∥x∥p + ∥y∥p
Setting p = 2 recovers the familiar Euclidean triangle inequality. Closely related is the Hölder inequality, which links two norms whose exponents are conjugate (1/p + 1/q = 1):
Σ |xᵢ yᵢ| ≤ ∥x∥p · ∥y∥q
Hölder's inequality is in fact one of the key ingredients used to prove Minkowski's inequality — which, in turn, is what makes the p-norm a genuine "distance" that obeys the triangle inequality. From three sticks on a table all the way up to abstract function spaces, it is the same simple idea at work.
FAQs
- What is the triangle inequality theorem?
- It states that three line segments form a triangle if and only if the sum of the lengths of every pair of sides is greater than the length of the remaining side: a + b > c, a + c > b, and b + c > a.
- Do 2, 3, and 5 form a triangle?
- No. Here 2 + 3 = 5, which is not greater than 5 — it is equal. The segments lie flat in a straight line, forming a degenerate triangle with zero area.
- Given two sides of 8 and 5, how long can the third side be?
- Using |a − b| < c < a + b, the third side must satisfy |8 − 5| < c < 8 + 5, i.e., 3 < c < 13. Any value strictly between 3 and 13 works.
- Does the triangle inequality only apply to right triangles?
- No. It holds for every triangle — acute, right, or obtuse. It is a rule about side lengths, not angles.
- What happens when a + b equals c exactly?
- The three points become collinear (they lie on a single straight line). This is a degenerate triangle: it has no area and no interior angles, so it is not considered a valid triangle.