Heronian Triangle — Definition, Formula & Examples

A Heronian triangle is a triangle whose side lengths and area are all positive integers. The simplest example is the well-known 3-4-5 right triangle, which has an area of 6.

A triangle with sides $a, b, c \in \mathbb{Z}^+$ is called Heronian if its area, computed via Heron's formula $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ , is also a positive integer.

Key Formula

A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}

Where:

$a, b, c$ = Integer side lengths of the triangle
$s$ = Semi-perimeter of the triangle
$A$ = Area, which must be a positive integer for a Heronian triangle

How It Works

To check whether a triangle is Heronian, first confirm that all three side lengths are positive integers and satisfy the triangle inequality. Then compute the semi-perimeter

s = \frac{a+b+c}{2}

and evaluate Heron's formula. If the result is a positive integer, the triangle is Heronian. Note that

s

does not need to be an integer itself — it just needs to produce an integer area. However, for

s

to be rational (and the formula to yield an integer),

a + b + c

must be even.

Worked Example

Problem: Determine whether the triangle with sides 13, 14, and 15 is Heronian.

Step 1: Compute the semi-perimeter.

s = \frac{13 + 14 + 15}{2} = 21

Step 2: Substitute into Heron's formula.

A = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}

Step 3: Simplify the product under the radical.

A = \sqrt{7056} = 84

Answer: The area is 84, a positive integer, so the 13-14-15 triangle is Heronian.

Why It Matters

Heronian triangles appear in number theory competitions and contest geometry where problems require all measurements to be integers. They also connect algebra (Heron's formula) with geometry, reinforcing how side lengths constrain area in non-obvious ways.

Common Mistakes

Mistake: Assuming every right triangle with integer sides is Heronian.

Correction: A Pythagorean triple like (3, 4, 5) gives area 6 (integer), so it works. But the concept applies to all triangles, not just right triangles — and not every integer-sided triangle has integer area.