Triangle Inequality Theorem Calculator
Check if three sides can form a triangle or find the possible range for a missing side
Triangle Inequality Check
Triangle Validity Check
Enter positive values for all three sides
Example Problems
Example 1: Valid Triangle
Given: Sides 3, 4, 5
Check: 3 + 4 = 7 > 5 ✓, 3 + 5 = 8 > 4 ✓, 4 + 5 = 9 > 3 ✓
Result: These sides can form a triangle (it's a right triangle!)
Example 2: Invalid Triangle
Given: Sides 1, 2, 4
Check: 1 + 2 = 3 ≤ 4 ✗
Result: These sides cannot form a triangle
Example 3: Finding Range
Given: Two sides of length 5
Range: |5 - 5| < c < 5 + 5 → 0 < c < 10
Result: Third side must be between 0 and 10 (exclusive)
Triangle Inequality Rules
Basic Rule
a + b > c, a + c > b, b + c > a
All three inequalities must be satisfied
Simplified Rule
Sum of two shortest > longest
Only need to check one inequality
Range Formula
|a - b| < c < a + b
For finding valid third side
Quick Reference
Valid Triangle
All inequalities satisfied
Invalid Triangle
One or more inequalities violated
Range Calculation
Find possible values for unknown side
Quick Check
Sum of shortest two > longest
Understanding the Triangle Inequality Theorem
What is the Triangle Inequality Theorem?
The Triangle Inequality Theorem states that for any triangle with sides a, b, and c, the sum of any two sides must be greater than the third side.
The Three Inequalities:
- a + b > c (sum of sides a and b is greater than c)
- a + c > b (sum of sides a and c is greater than b)
- b + c > a (sum of sides b and c is greater than a)
Important: ALL three inequalities must be satisfied for the sides to form a valid triangle.
Why Does This Matter?
Geometric Constraint
If one side is too long compared to the others, you can't "close" the triangle
Real-World Applications
Construction, engineering, navigation, and any field requiring triangular structures
Mathematical Foundation
Essential for trigonometry, coordinate geometry, and advanced mathematics
Simplified Checking Method
Instead of checking all three inequalities, you can use this simplified approach:
- Identify the longest side among the three
- Add the two shorter sides together
- If the sum is greater than the longest side, the triangle is valid
- If the sum is less than or equal to the longest side, the triangle is invalid