Triangle Inequality Calculator
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Triangle Inequality Tests
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Triangle Inequality Theorem
The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This must be true for all three combinations of sides.
The Three Conditions
For sides a, b, and c to form a valid triangle, ALL three conditions must be satisfied:
- a + b > c
- a + c > b
- b + c > a
If even one of these conditions fails, the three sides cannot form a triangle.
Why Does This Rule Exist?
Imagine trying to connect three sticks end-to-end to form a triangle. If one stick is too long (equal to or greater than the sum of the other two), the ends won't meet to close the triangle. The sides would just form a straight line or not connect at all.
Examples
Valid Triangle: 3, 4, 5
- 3 + 4 = 7 > 5 ✓
- 3 + 5 = 8 > 4 ✓
- 4 + 5 = 9 > 3 ✓
Invalid: 1, 2, 5
- 1 + 2 = 3 < 5 ✗ (Fails!)
- 1 + 5 = 6 > 2 ✓
- 2 + 5 = 7 > 1 ✓
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 must be true for all three combinations.
Do all three conditions need to pass?
Yes, all three conditions must be satisfied. If even one fails, the three sides cannot form a valid triangle.
What happens if a + b equals c?
If a + b = c (or any sum equals the third side), the sides form a degenerate triangle—a straight line, not a valid triangle.
Can I have sides 1, 2, and 10?
No, because 1 + 2 = 3, which is not greater than 10. The third side is too long relative to the other two.
What's the maximum length for the third side?
If you have sides a and b, the third side c must be less than a + b and greater than |a - b|.
Why is this theorem important?
It's fundamental to geometry and helps determine whether a triangle can exist before attempting more complex calculations. It's used in computer graphics, engineering, and navigation.
Does this work for all triangles?
Yes, the triangle inequality theorem applies to all triangles in Euclidean geometry: acute, right, obtuse, scalene, isosceles, and equilateral.
Can I use this with decimal numbers?
Yes, the triangle inequality theorem works with any positive real numbers, including decimals and fractions.