Heron's Formula Explained: Triangle Area From Three Sides Alone
How Heron's formula calculates a triangle's area from just its three side lengths, with no angle or height measurement needed, and a fully worked example.
Published July 20, 2026
Most triangle area formulas need a base and a height. Heron’s formula needs neither — just the three side lengths, which makes it the practical choice whenever height isn’t something you can easily measure directly.
The formula, in two steps
s = (a + b + c) ÷ 2
Step 1: find the semi-perimeter — half the triangle's total perimeter.
Area = √[s(s − a)(s − b)(s − c)]
Step 2: plug the semi-perimeter and all three sides into this single expression.
A fully worked example
For a triangle with sides 5, 6, and 7: the semi-perimeter is (5+6+7)/2 = 9. The area is √[9 × (9−5) × (9−6) × (9−7)] = √[9 × 4 × 3 × 2] = √216 ≈ 14.697 square units.
Heron's formula is named for Heron of Alexandria, who documented it around the 1st century AD — though there's some historical evidence the relationship may have been known earlier. It remains one of the oldest area formulas still in standard use today, essentially unchanged from its original form.
Why it works with only side lengths
Standard area formulas (½ × base × height) need a height — but height often isn't a measurement you have directly, especially for an irregular or non-right triangle.
No angle or height measurement is needed at all — three side lengths alone are mathematically sufficient to determine a triangle's area uniquely.
Three side lengths fully determine a triangle’s shape (this is the geometric principle called side-side-side congruence), which means the area is also fully determined by those three lengths — Heron’s formula is simply the algebraic expression of that fact, derived from combining the standard area formula with the Pythagorean theorem to eliminate the height term entirely.
When the three side lengths don’t form a valid triangle
| Condition | Result |
|---|---|
| Each side is shorter than the sum of the other two | Valid triangle, positive area |
| One side equals the sum of the other two | Degenerate (flat) triangle, area = 0 |
| One side is longer than the sum of the other two | Not a valid triangle at all |
This is the triangle inequality theorem, and it’s worth checking before applying Heron’s formula — plugging in side lengths that don’t satisfy it produces a negative number under the square root, which is exactly why a careful implementation (like the Triangle Area Calculator) clamps that value to prevent an invalid result rather than returning an error.
FAQ
Does Heron’s formula work for right triangles too? Yes — it works for any triangle regardless of angle type, including right triangles, where it will produce the identical result as the simpler ½ × base × height formula.
When should base-and-height be used instead of Heron’s formula? See Heron’s Formula vs. Base × Height: When to Use Which for a direct comparison of when each approach is more practical.
Can Heron’s formula handle decimal or very large side lengths? Yes — the formula works identically regardless of the units or magnitude of the side lengths, as long as all three use the same unit consistently.
Is there a version of Heron’s formula for other polygons? A generalization called Brahmagupta’s formula extends similar logic to cyclic quadrilaterals (four-sided shapes inscribed in a circle), though it requires additional conditions that don’t apply to arbitrary quadrilaterals the way Heron’s formula applies to any valid triangle.
Why does the formula use a square root? The area itself, when derived algebraically from the three side lengths, comes out as the square root of a product of four terms — this isn’t a simplification choice, it’s the direct algebraic result of the derivation.
How accurate is Heron’s formula compared to other area methods? It’s exact, not an approximation — given precise side length measurements, Heron’s formula produces the mathematically precise area, with any error coming only from the precision of the input measurements themselves.