Theorem

Triangle inequality

Geometry • Triangle • Triangle Inequality

Formula

a + b > c, b + c > a, c + a > b

For positive lengths

a

b

c

there is a triangle with those sides if and only if

a + b > c

b + c > a

and

c + a > b

; equivalently, the largest of the three is strictly less than the sum of the other two.

How to use it

1Sort the three lengths and test only the largest against the sum of the other two; the remaining two inequalities then hold automatically.
2Equality, as in $2 + 3 = 5$ , puts the three points on one line and encloses no area, so it is not a triangle.
3For the range of an unknown third side $x$ beside known sides $a$ and $b$ , the three inequalities collapse to $∣ a - b ∣ < x < a + b$ .

Worked example

For which values of

x

7

10

and

x

form a triangle?

Step-by-step

1
$x < 7 + 10 = 17$
Why: The unknown side must be less than the sum of the other two.
2
$7 + x > 10 \Rightarrow x > 3$
Why: Applying the inequality with $10$ as the largest side bounds $x$ from below.

Answer

3 < x < 17

Watch out

The inequalities are strict.

a + b = c

is the degenerate case in which the three points are collinear, so a triple such as

(2, 3, 5)

is not a triangle even though it only just fails.

Also known as

triangle inequality, sum of two sides greater than the third, do these lengths form a triangle, existence of a triangle from three lengths, range of the third side

Appears in past papers

Formula

Worked example

Appears in past papers

2020 (I)

2019 (I)

2018 (I)

2017 (II)

2017 (I)