Triangle Inequality Examples: 24 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Triangle Inequality.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Try to build a triangle with two short sticks and one very long one—you can't. The two short sticks can't reach across to close the shape. It's like trying to take a shortcut: the direct path (one side) is always shorter than going around (the other two sides combined).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A triangle exists only if any two sides can meet past the third side.

Common stuck point: The procedure for triangle inequality is the easy part; the trap is checking only one random pair. Asking "Is the sum of the two shorter sides greater than the longest side?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the sum of the two shorter sides greater than the longest side?

Worked Examples

Example 1

easy

Can sides of length 4, 7, and 10 form a triangle? Check all three triangle inequality conditions.

Check all three triangle inequality conditions for these sides.

Answer

Yes, sides 4, 7, and 10 can form a triangle.

First step

Step 1: The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side. Check all three combinations.

Full solution

2
Step 2: $4 + 7 = 11 > 10$ ✓
3
Step 3: $4 + 10 = 14 > 7$ ✓
4
Step 4: $7 + 10 = 17 > 4$ ✓ All three conditions are satisfied.

The Triangle Inequality requires all three pairwise sums to exceed the remaining side. In practice, only the sum of the two shortest sides needs to exceed the longest side — if that holds, the other two conditions are automatically satisfied. Here,

4 + 7 = 11 > 10

is the critical check.

Example 2

medium

Two sides of a triangle have lengths 8 and 13. Find the range of possible lengths for the third side.

Find the range of possible lengths for the third side x.

Example 3

medium

Two sides of a triangle measure

9

and

14

. List all integer values the third side could take.

Example 4

medium

A triangle has sides

a

a + 3

, and

a + 5

. Find the smallest integer value of

a

for which the triangle exists.

Example 5

medium

Can a triangle have sides

\sqrt{2}, \sqrt{3}, \sqrt{8}

Example 6

medium

Two sides of a triangle measure

13

and

20

. The third side is a multiple of

4

. List all possible third-side lengths.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Which of these sets of side lengths cannot form a triangle? (a) 3, 4, 5. (b) 1, 2, 3. (c) 5, 8, 12.

Example 2

hard

\triangle ABC

AB = 6

and

BC = 10

. The perimeter is 24. Is this a valid triangle? Find

AC

and verify.

Example 3

easy

Can sides of length

6

9

, and

12

form a triangle? Check the inequalities.

Check all three triangle inequality conditions.

Example 4

easy

Can sides

1

1

, and

3

form a triangle?

Example 5

medium

Which of these triples form triangles? (a)

5, 12, 13

; (b)

7, 8, 16

; (c)

10, 10, 10

Example 6

easy

Three sides of lengths

3, 4, 7

— does this form a triangle?

Example 7

medium

If two sides of a triangle are

a

and

b

with

a \geq b

, what is the range for the third side

c

Example 8

medium

Sides of a triangle are

x, x + 2, x + 4

. What positive integer values of

x

are valid?

Example 9

hard

The perimeter of a triangle is

30

. Two sides are

11

and

12

. Is the triangle valid? What is the third side?

Perimeter is 30; find the third side x and verify the triangle is valid.

Example 10

easy

Two sides of a triangle are

2

and

9

. Which of

7

8

11

could be the third side?

Example 11

hard

A triangle has integer side lengths and perimeter

15

. If one side is

6

, find the number of possible triangles.

Example 12

medium

Three sticks measure

8, 11, 19

. Will they form a triangle if you bend them flat together?

Example 13

hard

Prove using triangle inequalities that for any triangle with sides

a, b, c

|a - b| < c < a + b

Example 14

medium

Sides of a triangle are

a = 2x + 1

b = 3x - 2

c = x + 5

. For what range of

x

is the triangle valid?

Example 15

easy

Three sticks of length

3

cm,

3

cm, and

5

cm. Can you form a triangle?

Can sides 3, 3, and 5 form a triangle? Verify with the inequality.

Example 16

challenge

An isosceles triangle has two equal sides of length

10

and an integer third side. How many such triangles are possible?

Example 17

medium

A triangle has sides

7

24

x

and is a right triangle. Use the triangle inequality and Pythagoras to find

x

Example 18

hard

Three positive real numbers

a, b, c

satisfy

a^2 + b^2 + c^2 = 50

and

a + b + c = 12

. Could they form a triangle?

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Related Concepts

Triangles Addition Comparison Pythagorean Theorem

Background Knowledge

These ideas may be useful before you work through the harder examples.

trianglesadditioncomparison