How it works
Three lengths form a triangle exactly when each one is less than the sum of the other two. In practice you only need to check that the longest side is shorter than the other two combined.
If two sides are a and b with a ≤ b, the third side x must satisfy
b − a < x < a + b
The inequalities are strict: if x equals a + b, the "triangle" collapses into a line. For whole-number sides there are 2a − 1 possible values. Watch the isosceles trap: with sides 7 and 14, the third side must be 14, because 7 + 7 is not more than 14.
Triangle Inequality practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Two sides of a triangle are 4 and 9. Find the smallest integer possible for the third side.
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Answer: 6
The third side must be more than 9 − 4 = 5, so the smallest whole number is 6.
An isosceles triangle has two sides 7 and 14. Find its perimeter.
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Answer: 35
The sides are either 7, 7, 14 or 7, 14, 14. The first fails because 7 + 7 = 14 is not greater than 14. So the sides are 7, 14, 14 and the perimeter is 35.
How many integer lengths can the third side take with sides 7 and 10?
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Answer: 13
The third side is between 10 − 7 = 3 and 10 + 7 = 17, not inclusive: 4 through 16, which is 13 values.
Two sides of a triangle are 12 and 25. How many odd integer lengths are possible for the third side.
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Answer: 11
The third side x satisfies 13 < x < 37. The odd values are 15, 17, …, 35, which is (35 − 15) ÷ 2 + 1 = 11 values.
Keep going
Related topics
- Pythagorean theorem
- Polygon interior and exterior angle
- Triangle area
- Special right triangles: 45-45-90 and 30-60-90
Related practice sets in the app
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice