Check three sides
Use this when all side lengths are known.
Valid triangle, invalid triangle, or degenerate flat boundary.
All three strict inequalities must pass.
Triangle Inequality Theorem Calculator
Check if three sides can form a triangle — or find the possible range of the third side, with a visual drag-and-check demo.
can these sides form a trianglefind range of third side of triangletriangle inequality theorem examples
Geometry first
This page is the geometry bridge in the inequality series: it checks side lengths, computes an open interval for the unknown side, and keeps the visual proof lightweight with SVG instead of a geometry engine.
Two-mode calculator
Check sides or find the third-side range
Yes. All three strict inequalities pass, so these sides can form a triangle.
a + b > c
a + b > c → 12 > 10
a + c > b
a + c > b → 15 > 7
b + c > a
b + c > a → 17 > 5
Step-by-step breakdown
- Test all three strict inequalities: a + b > c, a + c > b, and b + c > a.
- a + b > c → 12 > 10
- a + c > b → 15 > 7
- b + c > a → 17 > 5
- Because all three conditions are true, the line segments close into a triangle.
Drag-and-check triangle
Closed triangle
These sides close into a triangle because every strict inequality passes.
side a5
side b7
side c10
Visual theorem
The three possible states: closed, broken, or flat
Closed
All three sums are strictly greater than the remaining side, so the triangle encloses area.
Broken
One side is too long. The shorter two cannot reach far enough to close the shape.
Flat
A boundary equality such as 3 + 4 = 7 creates a straight line, not a triangle.
Theorem
What the triangle inequality theorem checks
The triangle inequality theorem says that any two sides of a triangle must add up to more than the third side. The word any matters: all three pairwise checks must pass before three segments can close into one triangle.
The reverse problem is just as useful. If two sides are known, the third side must be longer than the positive gap between them and shorter than their total. That is why the answer is an open interval: |a - b| < c < a + b.
Equality is the boundary case. If a + b = c, the segments can lie end-to-end, but they have no height and no area. This is the geometric reason the theorem uses a strict greater-than sign instead of greater-than-or-equal-to.
a + b > c
must be true
a + c > b
must be true
b + c > a
must be true
Mode Guide
Choose the right triangle inequality workflow
Find the third-side range
Use this when two side lengths are known and one side is variable.
An open interval such as 3 < c < 17.
The lower endpoint is |a - b| and the upper endpoint is a + b.
Interpret the endpoints
Use this before writing interval notation.
Both endpoints are excluded because equality gives a flat line.
Triangle side ranges use parentheses, not brackets.
Common Mistakes
Boundary cases that look valid but are not triangles
Checking only the two smallest sides against the largest side
That shortcut works after sorting, but the calculator still displays all three checks so the reasoning is auditable.
Including the third-side endpoints
Use strict inequalities. The endpoints make a flattened shape, not a triangle with area.
Allowing zero or negative side lengths
Every side length must be positive before the triangle inequality theorem is even tested.
Examples
Triangle inequality theorem examples
Valid triangle
5, 7, 10
5 + 7 = 12 > 10, 5 + 10 = 15 > 7, and 7 + 10 = 17 > 5. All three pass.
Invalid triangle
3, 4, 8
3 + 4 = 7, and 7 < 8. Even though two conditions pass, ALL THREE must be satisfied.
Degenerate boundary
3, 4, 7
3 + 4 = 7 exactly, so the triangle flattens into a straight line with no area.
Third side range
7, 10
|7 - 10| < c < 7 + 10, so 3 < c < 17.
Two equal known sides
5, 5
|5 - 5| < c < 5 + 5, so 0 < c < 10.
Geometric Inequalities
A geometry entry point for the inequality toolkit
The third-side range is technically a compound inequality, and its answer is naturally written in interval notation. Use these connected calculators when you want to translate the geometry result back into algebra notation.
Interval Notation Calculator
Translate the third-side answer (3, 17) into inequality notation and a number line.
Compound Inequality Calculator
Read |a - b| < c < a + b as an AND compound inequality.
Graphing Inequalities Calculator
Move from geometric side constraints to coordinate-plane inequality graphs.
FAQ
Frequently Asked Questions
What is the triangle inequality theorem?
The triangle inequality theorem says that the sum of any two side lengths in a triangle must be greater than the third side. For sides a, b, and c, the required checks are a + b > c, a + c > b, and b + c > a.
Why must the sum of two sides be greater than the third side?
If two sides do not add up to more than the third side, their endpoints cannot meet to enclose area. They either fall short of closing or lie flat along the longest side.
Can two sides of a triangle be equal to the third side?
No. Equality creates a degenerate case, not a real triangle. For example, 3, 4, and 7 flatten into a straight line because 3 + 4 = 7.
How do you find the range of possible values for the third side?
If the two known sides are a and b, the third side c must satisfy |a - b| < c < a + b. The lower endpoint is excluded, and the upper endpoint is excluded.
Does the triangle inequality theorem apply to all types of triangles?
Yes. Scalene, isosceles, equilateral, acute, right, and obtuse triangles all must satisfy the same three triangle inequality conditions.
What happens if the sides exactly satisfy a + b = c?
The shape becomes a straight-line boundary case with zero area. It is often called degenerate, and it is why a valid triangle requires a strict inequality.