How this instrument works
Heron's Formula finds a triangle's area from its three side lengths alone, with no angle or height needing to be measured or computed separately. It first computes the semi-perimeter, s = (a+b+c) ⁄ 2, half the triangle's total perimeter, then the area is the square root of s multiplied by each side's own 'deficit' from s: A = √(s(s−a)(s−b)(s−c)).
The formula is named for Heron of Alexandria, who documented it roughly two thousand years ago, though evidence suggests the result may have been known even earlier. Its real strength is generality: it works identically for a right triangle, an obtuse triangle, or a wildly scalene one, without ever needing to identify which side plays which role or drop a perpendicular height by hand.
For the formula to produce a real, positive area, the three side lengths must actually satisfy the Triangle Inequality — each side shorter than the sum of the other two. If that condition fails, at least one of the deficit terms (s−a), (s−b), or (s−c) turns negative, and the product under the square root goes negative too, correctly signaling that no real triangle exists with those three side lengths.
- Enter the triangle's three side lengths into the Side a, Side b, and Side c fields.
- Read Area: the sheet computes the semi-perimeter first, then applies Heron's Formula.
- The three sides must satisfy the Triangle Inequality (any two sides must sum to more than the third), or the sheet will flag the input as invalid.
Worked example — sides 3, 4, and 5
A triangle has sides 3, 4, and 5. Its semi-perimeter is s = (3+4+5) ⁄ 2 = 6, and its area is √(6 × (6−3) × (6−4) × (6−5)) = √(6×3×2×1) = √36 = 6 exactly — matching what the plain ½·base·height formula would give for this right triangle, confirming Heron's Formula agrees with simpler methods whenever they're also available.
A less obviously special triangle with sides 13, 14, and 15 has semi-perimeter s = 21, giving an area of √(21×8×7×6) = √7056 = 84 exactly — a clean whole-number result despite the triangle having no special symmetry, right angle, or equal sides at all.
Questions
What is Heron's Formula?
A = √(s(s−a)(s−b)(s−c)), where a, b, c are a triangle's three side lengths and s = (a+b+c)⁄2 is the semi-perimeter. It finds any triangle's area from its three sides alone, with no angle or height needing to be known separately.
Does Heron's Formula work for any triangle?
Yes — it works identically for right, acute, obtuse, and scalene triangles, since it never assumes a particular angle or requires identifying a base and matching height. Only the three side lengths themselves matter.
What happens if the three sides can't form a real triangle?
One of the deficit terms, (s−a), (s−b), or (s−c), turns negative, making the product under the square root negative and giving an undefined (non-real) result — the formula's own way of signaling that those three lengths violate the Triangle Inequality and can't form a genuine triangle.
Who discovered Heron's Formula?
It's named for Heron of Alexandria, who documented the result roughly two thousand years ago, though some historians believe it may have been known earlier still, possibly to Archimedes.
Does Heron's Formula still work for an equilateral triangle?
Yes — plugging in three equal sides gives the exact same result as the dedicated equilateral-triangle area formula, (√3⁄4)s², confirming Heron's Formula is fully general and needs no separate handling for special triangle types.