Triangle Inequality Theorem Calculator

Enter three side lengths — Side A, Side B, and Side C — into the Triangle Inequality Theorem Calculator to find out whether those measurements can form a valid triangle. You'll get an instant validity verdict along with all three inequality checks (a + b > c, a + c > b, b + c > a), so you can see exactly which conditions pass or fail.

Enter the length of the first side

Enter the length of the second side

Enter the length of the third side

Results

Triangle Valid?

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A + B > C

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A + C > B

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B + C > A

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Min Possible Third Side (given A & B)

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Max Possible Third Side (given A & B)

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Side Lengths vs. Triangle Inequality Conditions

Results Table

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 greater than the third side. This rule applies to all three combinations: a + b > c, a + c > b, and b + c > a. If any one of these conditions fails, the three lengths cannot form a valid triangle.

How do I check if three lengths make a triangle?

To verify three side lengths form a triangle, test all three inequality conditions: add each pair of sides and confirm the sum is strictly greater than the remaining side. If all three checks pass, a valid triangle can be formed. This calculator does all three checks for you automatically.

Do the sides 4, 5, and 10 make a triangle?

No — sides 4, 5, and 10 do not form a valid triangle. The condition 4 + 5 > 10 simplifies to 9 > 10, which is false. Since at least one inequality condition fails, these three lengths cannot be the sides of any triangle.

What is the third side of a triangle with two sides equal to 5?

If two sides are both 5, the third side must satisfy the inequalities: it must be greater than 5 + 5 − 5 = 0 (essentially greater than 0) and less than 5 + 5 = 10. So the third side can be any length strictly between 0 and 10 (exclusive).

Are all side length combinations possible in a triangle?

No. Any set of three positive lengths must satisfy all three triangle inequality conditions to be valid. For example, very small sides paired with a very long side will fail because the two smaller sides together won't exceed the longest side.

What happens if one of the inequality conditions is exactly equal (e.g. a + b = c)?

If the sum of two sides equals the third exactly, the three points would be collinear — they would form a straight line rather than a closed triangle. This is called a degenerate triangle, and it is not considered a valid triangle. All three inequalities must be strict (greater than, not equal to).

How do I find the range of a possible third side given two known sides?

If you know two sides, say lengths a and b, the third side c must satisfy: |a − b| < c < a + b. The minimum possible value is just above the absolute difference of the two sides, and the maximum is just below their sum. This calculator displays both boundary values for your given Side A and Side B.

Can the Triangle Inequality Theorem be applied to non-Euclidean geometry?

The standard Triangle Inequality Theorem applies to Euclidean (flat) geometry. In non-Euclidean geometries such as spherical or hyperbolic geometry, modified versions of the inequality hold, but they differ in their exact form. For everyday geometry problems, the Euclidean version used in this calculator is the correct one to apply.

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