Triangle Inequality Theorem Calculator

Triangle Inequality Theorem

Three positive lengths $a$, $b$, and $c$ form a triangle only if each pair sums to more than the third side:

$$a + b > c, \quad a + c > b, \quad b + c > a$$

Equivalently, the sum of the two shortest sides must exceed the longest side. See also: 3 Sides Triangle Area Calculator, Classifying Triangles Calculator, and Triangle Calculator.

Worked Example

Sides 3, 4, and 10 fail because $3 + 4 = 7 \leq 10$. Sides 5, 7, and 10 pass all three checks and form a valid triangle with area about 17.32 using Heron's formula.

Frequently Asked Questions

Can any three positive numbers be triangle sides?

No. If the longest side is greater than or equal to the sum of the other two, no triangle exists.

What is the shortcut rule?

Sort the three lengths. They form a triangle if and only if the sum of the two smallest exceeds the largest.

Do 4, 5, and 10 make a triangle?

No. $4 + 5 = 9$, which is less than 10, so the triangle inequality fails.

What happens when the inequality passes?

The calculator also reports semi-perimeter and area using Heron's formula so you can continue solving geometry problems.

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Triangle Inequality Theorem Calculator