Triangle Inequality Theorem Calculator

Enter three side lengths to check if they form a valid triangle, with all inequality conditions shown and full triangle properties calculated.

📐 Triangle Inequality Theorem Calculator
Side a3
units
0.1100
Side b4
units
0.1100
Side c5
units
0.1100
Valid Triangle?
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📐 What is the Triangle Inequality Theorem?

The triangle inequality theorem states that the sum of any two sides of a triangle must be strictly greater than the third side. For three side lengths a, b, and c to form a valid triangle, all three conditions must hold simultaneously: a + b > c, a + c > b, and b + c > a. If even one condition fails, the three lengths cannot form a triangle in Euclidean geometry, regardless of how close the numbers are to satisfying the inequality.

This theorem is foundational in geometry, trigonometry, and real-world measurement. Builders use it to verify whether three given lengths can frame a triangular structure. Surveyors apply it when triangulating land boundaries. Physicists use it to describe when three force vectors can achieve equilibrium in a closed triangle. Students encounter it when checking whether a proposed triangle with integer side lengths is valid before computing area or angles.

A common misconception is that only the largest side needs to be checked against the other two. While it is true that if the largest side is less than the sum of the other two, all three conditions are automatically satisfied, verifying all three conditions explicitly is useful for teaching and understanding why the theorem works. This calculator shows each condition individually with a clear pass or fail label and the actual arithmetic.

Beyond the simple validity check, this calculator also computes full triangle properties when the sides are valid: area via Heron's formula, perimeter, all three interior angles via the Law of Cosines, and the triangle type by both sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). This makes it a complete tool for quickly solving any triangle when all three sides are known, not just a validity checker.

📐 Formula

a + b > c     a + c > b     b + c > a
a, b, c = the three side lengths of the proposed triangle
All three conditions must hold for a valid triangle
Shortcut: Sort the sides so c is largest. Check only a + b > c.
Area = √(s(s − a)(s − b)(s − c))    s = (a + b + c) ÷ 2
s = semi-perimeter of the triangle
cos A = (b² + c² − a²) ÷ (2bc)  [Law of Cosines]
Example: Sides 3, 4, 5. s = 6. Area = √(6 × 3 × 2 × 1) = 6 sq units.

📖 How to Use This Calculator

Steps

1
Enter side lengths - Type or drag the sliders to set positive values for sides a, b, and c. All three sides must use the same unit of measurement.
2
Click Calculate - Press Calculate to see all three inequality conditions checked instantly, each with a green checkmark for pass or red cross for fail.
3
Read the verdict and properties - If all three conditions pass, the calculator shows area, perimeter, all three angles, and the triangle classification. If any condition fails, only the verdict and the specific failing conditions are shown.

💡 Example Calculations

Example 1 - Valid scalene triangle: 5, 7, 10

Check if sides 5, 7, 10 form a valid triangle

1
Check condition 1: 5 + 7 = 12, which is greater than 10. Pass.
2
Check condition 2: 5 + 10 = 15, which is greater than 7. Pass. Check condition 3: 7 + 10 = 17, which is greater than 5. Pass.
3
All conditions pass. Compute area: s = (5 + 7 + 10) / 2 = 11. Area = sqrt(11 times 6 times 4 times 1) = sqrt(264) = 16.25 sq units. Type: Scalene, Obtuse (largest angle opposite side 10 is greater than 90 degrees).
Result: Valid triangle. Area = 16.25 sq units, Perimeter = 22 units
Try this example →

Example 2 - Invalid triangle: 2, 3, 8

Check if sides 2, 3, 8 can form a triangle

1
Check condition 1: 2 + 3 = 5. Is 5 greater than 8? No. This condition fails.
2
The remaining conditions: 2 + 8 = 10 greater than 3 (pass), 3 + 8 = 11 greater than 2 (pass). But one condition already fails.
3
Because condition 1 fails, these sides cannot form a triangle. The two shorter sides (2 and 3) cannot stretch far enough to close the triangle with the longest side (8).
Result: Not a valid triangle. Condition a + b > c fails (5 is not greater than 8)
Try this example →

Example 3 - Equilateral triangle: 6, 6, 6

Verify and compute properties for an equilateral triangle with side 6

1
Check all conditions: 6 + 6 = 12 greater than 6 (pass), for all three pairings. All pass with maximum margin.
2
Area by Heron's formula: s = 9. Area = sqrt(9 times 3 times 3 times 3) = sqrt(243) = 15.59 sq units. Alternatively, A = (sqrt(3) / 4) times 36 = 15.59.
3
All three angles = 60 degrees (equilateral). Triangle type: Equilateral, Acute. Perimeter = 18 units.
Result: Valid equilateral triangle. All angles 60 degrees, area = 15.59 sq units
Try this example →

❓ Frequently Asked Questions

What is the triangle inequality theorem?+
The triangle inequality theorem states that the sum of any two sides of a triangle must be strictly greater than the third side. For sides a, b, c, all three conditions must hold: a + b greater than c, a + c greater than b, and b + c greater than a. If any one condition fails, the three lengths cannot form a valid triangle in Euclidean geometry.
Can 1, 2, 3 form a triangle?+
No. The critical check is 1 + 2 = 3, but the theorem requires strictly greater than, not equal to. Because 1 + 2 is not greater than 3 (it equals 3), the three sides would be collinear and form a degenerate triangle with zero area. The condition a + b greater than c must be a strict inequality. Sides 1, 2, 3 form a degenerate (flat) figure, not a triangle.
Can 3, 4, 5 form a triangle?+
Yes. Check: 3 + 4 = 7 greater than 5 (pass), 3 + 5 = 8 greater than 4 (pass), 4 + 5 = 9 greater than 3 (pass). All three conditions hold, so 3-4-5 forms a valid triangle. It is also a right triangle because 3 squared + 4 squared = 9 + 16 = 25 = 5 squared. Area = 6 square units, angles = 90, 53.13, 36.87 degrees.
Why does only the largest side need to be checked?+
If you sort the sides so c is largest, the conditions a + b greater than c, a + c greater than b, and b + c greater than a simplify. Because c is already largest, a + c is guaranteed greater than b (since c alone is already at least as large as b, and a is positive). Similarly b + c is guaranteed greater than a. So only a + b greater than c needs to be verified. This calculator still shows all three for educational completeness.
What are the three conditions of the triangle inequality?+
The three conditions for sides a, b, c are: (1) a + b is greater than c, (2) a + c is greater than b, (3) b + c is greater than a. Each condition checks whether one side is too long relative to the other two. All three must hold simultaneously. This calculator displays each condition with the actual arithmetic and a pass or fail indicator.
Can 5, 12, 13 form a triangle?+
Yes. Checking: 5 + 12 = 17 greater than 13 (pass), 5 + 13 = 18 greater than 12 (pass), 12 + 13 = 25 greater than 5 (pass). This is also a right triangle because 5 squared + 12 squared = 25 + 144 = 169 = 13 squared. It is a Pythagorean triple. Area = (5 times 12) / 2 = 30 square units.
What is a degenerate triangle?+
A degenerate triangle is a set of three side lengths where the sum of two sides exactly equals the third (for example, 1-2-3 or 2-4-6). The three points would be collinear, the enclosed area would be zero, and no actual triangular region would exist. The triangle inequality requires the sum to be strictly greater than (not equal to) the third side, so degenerate cases fail the strict version of the theorem.
How does this calculator determine triangle type?+
Classification by sides: equilateral if all three sides are equal, isosceles if exactly two sides are equal, scalene if all three sides differ. Classification by angles: the calculator computes all three angles using the Law of Cosines, then checks the largest angle. Acute if all angles are less than 90 degrees, right if the largest angle equals exactly 90 degrees, obtuse if the largest angle exceeds 90 degrees.
What formula does this calculator use for area?+
This calculator uses Heron's formula: Area = square root of (s times (s minus a) times (s minus b) times (s minus c)), where s = (a + b + c) / 2 is the semi-perimeter. This formula works for any triangle given all three sides, with no need for angles or height. For 5-7-10: s = 11, Area = sqrt(11 times 6 times 4 times 1) = sqrt(264) = 16.25 square units.
How is the triangle inequality used in coordinate geometry?+
In coordinate geometry, the triangle inequality guarantees that the straight-line distance between two points is always shorter than or equal to any path through a third point. Formally, for points P, Q, R: distance(P, R) is less than or equal to distance(P, Q) plus distance(Q, R). This is the basis for many optimization and pathfinding algorithms, and it defines what a metric space is in advanced mathematics.
Can the triangle inequality be applied to obtuse triangles?+
Yes, the triangle inequality applies to all triangles regardless of angle type. An obtuse triangle like 5-7-10 (where the angle opposite side 10 is greater than 90 degrees) still satisfies 5 + 7 = 12, which is greater than 10, so the triangle is valid. The triangle inequality is a geometric condition on side lengths only and does not depend on the angle measures.
Tags: Triangle Inequality Theorem Calculator Can Three Sides Form a Triangle Triangle Inequality Valid Triangle Checker Triangle Sides Calculator Calculator Formula Free Online

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What is the triangle inequality theorem?

The triangle inequality theorem states that the sum of any two sides of a triangle must be strictly greater than the third side. For three sides a, b, and c, all three conditions must hold: a + b > c, a + c > b, and b + c > a. If any condition fails, the sides cannot form a triangle. This is a necessary and sufficient condition for a valid triangle.

How do you check if three sides form a triangle?

Sort the three numbers from smallest to largest. Call them p, q, r where r is largest. The only condition you need to check is p + q > r, because the other two conditions are automatically satisfied when r is the largest. If p + q > r, the sides form a valid triangle. Example: check 5, 7, 10. Smallest two sum: 5 + 7 = 12 > 10. Valid triangle.

Can 3, 4, 8 form a triangle?

No. Check the critical condition: 3 + 4 = 7, which is not greater than 8 (7 is less than 8). The triangle inequality fails for condition a + b > c. These three lengths cannot form any triangle. The gap is 1 unit, meaning the two shorter sides fall exactly 1 unit short of reaching each other when the longest side is straight.

Can 5, 12, 13 form a triangle?

Yes. Check all three conditions: 5 + 12 = 17 > 13 (pass), 5 + 13 = 18 > 12 (pass), 12 + 13 = 25 > 5 (pass). All three inequalities hold. Moreover, 5 squared + 12 squared = 25 + 144 = 169 = 13 squared, so this is a right triangle and a Pythagorean triple.

What is a degenerate triangle?

A degenerate triangle occurs when the three sides satisfy exactly a + b = c for some arrangement (for example, sides 1, 2, 3: 1 + 2 = 3 exactly). The three points would be collinear, the area would be zero, and there would be no enclosed region. A degenerate triangle fails the strict triangle inequality (it needs strictly greater than, not equal). This calculator requires a + b > c, not a + b greater than or equal to c.

What triangle types does this calculator identify?

This calculator classifies valid triangles by two criteria. By sides: equilateral (all three sides equal), isosceles (exactly two sides equal), or scalene (all three sides different). By angles: acute (all angles less than 90 degrees), right (one angle equals exactly 90 degrees), or obtuse (one angle greater than 90 degrees). Both labels are shown simultaneously, for example Isosceles, Acute.

How does Heron's formula calculate triangle area?

Heron's formula computes triangle area from the three side lengths without needing angles or height. First compute the semi-perimeter s = (a + b + c) divided by 2. Then area = square root of (s times (s minus a) times (s minus b) times (s minus c)). Example: sides 3, 4, 5. s = 6. Area = sqrt(6 times 3 times 2 times 1) = sqrt(36) = 6 square units.

Why do we need all three conditions for the triangle inequality?

Technically, if you already know which side is longest, you only need to check one condition. But in general, without sorting, you must check all three: a + b > c, a + c > b, and b + c > a. This calculator checks all three and displays each result separately so you can see exactly which condition passes or fails, which is useful for educational understanding.

What angle does this calculator use the Law of Cosines for?

For a valid triangle with sides a, b, c, the calculator computes all three angles using the Law of Cosines: angle A = arccos((b squared + c squared minus a squared) divided by (2bc)), and similarly for B and C. The three angles are then classified as acute, right, or obtuse based on the largest angle. Angles are shown in decimal degrees.

Can the triangle inequality theorem be used in 3D geometry?

Yes. The triangle inequality extends to any metric space, meaning the concept applies to distances in 3D coordinates, graph edges, and abstract spaces. In 3D geometry, if three points are given by coordinates, you can compute the three distances between them and then check the triangle inequality to confirm they form a non-degenerate triangle.

What happens if two sides are equal in this calculator?

If exactly two sides are equal, the triangle is classified as isosceles. All three inequality conditions are still checked and displayed. An isosceles triangle with sides a, a, b is always valid as long as 2a > b and b > 0. The degenerate case a + a = b (where b = 2a) would fail the strict inequality and be flagged as invalid.

What is the relationship between the triangle inequality and the Pythagorean theorem?

The triangle inequality is a necessary condition for any triangle. The Pythagorean theorem is an additional equality condition that holds only for right triangles. A right triangle with legs a, b and hypotenuse c satisfies both: a + b > c (triangle inequality) AND a squared + b squared = c squared (Pythagorean). All Pythagorean triples form valid triangles that satisfy the triangle inequality.

📌 Quick Tips

💡The triangle inequality theorem is symmetric. If any one of the three conditions fails, the sides cannot form a triangle. You only need to find one violation to confirm it is invalid.
💡An equilateral triangle (all sides equal) always satisfies the triangle inequality with maximum margin. Each sum of two sides equals twice the third side, which is always greater.
💡When two sides are very different in length, check whether the longer side is less than the sum of the other two. A side of 10 with two sides of 1 and 2 fails because 1 + 2 = 3, which is not greater than 10.
💡Right triangles satisfy the triangle inequality and also satisfy a squared plus b squared equals c squared for the hypotenuse c. Use the Pythagorean Theorem Calculator to check right-triangle conditions.
💡Degenerate triangles occur when the sum of two sides exactly equals the third (e.g., 1-2-3: 1 + 2 = 3, not strictly greater). The area would be zero. This calculator requires strict inequality.