Triangle Inequality Theorem
Check if three side lengths satisfy the triangle inequality theorem. Enter sides a, b, c to verify a+b>c, a+c>b, and b+c>a for instant geometry results.
About This Calculator
About the Triangle Inequality Theorem Calculator
The triangle inequality theorem is one of the most fundamental principles in geometry. It states that three side lengths can form a triangle if and only if the sum of any two sides is greater than the third side. This free triangle inequality theorem calculator instantly checks all three conditions -- a + b > c, a + c > b, and b + c > a -- and tells you whether your three side lengths make a valid triangle.
Who is this calculator for?
This tool is designed for students learning geometry, teachers preparing lesson materials, engineers verifying structural dimensions, carpenters checking wood lengths, and anyone who needs to quickly determine if three given lengths can form a triangle. It is especially useful for geometry homework, exam preparation, and real-world construction planning.
How the triangle inequality theorem works
For three positive side lengths a, b, and c to form a triangle, three conditions must hold simultaneously:
- a + b > c -- the sum of the two smaller sides must exceed the largest
- a + c > b -- the sum of side a and side c must exceed side b
- b + c > a -- the sum of side b and side c must exceed side a
If all three inequalities are satisfied, you have a valid triangle. If any inequality fails, those three lengths cannot form a triangle. For example, sides 3, 4, and 5 satisfy all conditions (3+4=7>5, 3+5=8>4, 4+5=9>3), while sides 1, 2, and 3 do not because 1+2=3 equals but does not exceed 3.
The triangle inequality theorem also has important generalizations in higher mathematics, including the triangle inequality with absolute values (|a+b| <= |a| + |b|), the reverse triangle inequality (||a|-|b|| <= |a-b|), and the Minkowski and Hölder inequalities used in advanced analysis and functional spaces.
Frequently Asked Questions
What is the triangle inequality theorem?
The triangle inequality theorem states that for any three side lengths to form a triangle, the sum of any two sides must be greater than the third side. In other words, a + b > c, a + c > b, and b + c > a must all be true simultaneously.
How do I check if three side lengths make a triangle?
To check if three side lengths make a triangle, verify all three inequalities: a + b > c, a + c > b, and b + c > a. If all three hold true, the sides form a valid triangle. If any inequality fails, the sides cannot form a triangle. For example, sides 3, 4, and 5 satisfy all three inequalities (3+4>5, 3+5>4, 4+5>3) so they form a valid triangle.
Do the sides 1, 2, and 3 form a triangle?
No, sides 1, 2, and 3 do not form a triangle. This is because 1 + 2 = 3, which is equal to the third side but not greater than it. The triangle inequality requires a strict greater-than, not greater-than-or-equal, so these lengths fail the theorem.
What is the reverse triangle inequality?
The reverse triangle inequality states that ||a| - |b|| <= |a - b| for any two numbers. For triangles specifically, this means that the difference between the lengths of any two sides is always less than or equal to the length of the third side. It is a useful corollary of the standard triangle inequality.
Can a triangle have sides with lengths that are decimals or fractions?
Yes, triangles can have side lengths expressed as decimals or fractions. The triangle inequality theorem applies to any positive real numbers, not just integers. For example, sides 2.5, 3.7, and 5.1 can form a triangle if all three inequalities are satisfied.
What happens if I enter a negative number or zero?
Side lengths must be positive numbers greater than zero. If you enter a negative number or zero, the calculator will not compute a result -- it will prompt you to enter valid positive side lengths. Triangles cannot have zero or negative side lengths by definition.
Is this triangle inequality calculator free to use?
Yes, this triangle inequality calculator is completely free to use with no registration or hidden fees. You can check unlimited triangle side combinations instantly.