🔻 Triangle Area Calculator (Heron's Formula)
Sides 3, 5, 6 → area = 2√14 exactly — via Heron's formula in its integer-safe form, with the triangle inequality checked and every step shown.
Sides accept integers, decimals and fractions. The triangle inequality is checked before computing.
A = 2√14
≈ 7.4833148
area (exact simplified radical)
14
perimeter
7
semi-perimeter s
Working (Heron's formula, integer-safe form)
- 16A² = (a+b+c)(−a+b+c)(a−b+c)(a+b−c) — Heron's formula with the fractions cleared
- 16A² = (14)(8)(4)(2) = 896
- A = √(896) / 4 = 2√14 — simplified radical form
Exact areas: sides 3, 5, 6 give exactly 2√14, and 3-4-5 gives exactly 6 — where every decimal calculator prints 7.4833…. Runs locally.
How the triangle area calculator (heron's formula) works
Heron's formula computes a triangle's area from its three sides alone. The tool uses the fraction-free form 16A² = (a+b+c)(−a+b+c)(a−b+c)(a+b−c): for rational sides that product is an exact rational, so the area is an exact square root — which the tool simplifies symbolically, giving 2√14 for sides 3-5-6 and exactly 6 for the 3-4-5 right triangle (a Heronian triangle, where the area comes out whole). The triangle inequality is verified first, with a clear message when the three lengths simply cannot form a triangle. Perimeter and semi-perimeter are shown alongside, and every step of the computation is written out.
The exact radical is the differentiator — every mainstream calculator answers 7.4833 for sides 3-5-6, but "2√14" is what the geometry answer key says, and A² = 56 is what the next line of the problem often needs. The 16A² form also deserves its footnote: computing Heron's formula through the semi-perimeter in floating point famously loses precision for thin triangles, a numerical-analysis classic that exact arithmetic sidesteps entirely.
Frequently asked questions
What is Heron's formula?
Area = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter (a+b+c)/2 — the area from sides alone, no heights or angles needed. The tool uses the equivalent fraction-free form 16A² = (a+b+c)(−a+b+c)(a−b+c)(a+b−c).
Why is the answer a square root?
Heron's formula produces A², so the area is its square root — exact only if kept symbolic. Sides 3, 5, 6 give 16A² = 896, so A = √896/4 = 2√14. The tool simplifies the radical automatically.
What is a Heronian triangle?
One with integer sides AND integer area — like 3-4-5 (area 6) or 5-5-6 (area 12). The tool celebrates when your triangle turns out to be one.
What if my three lengths can't form a triangle?
The triangle inequality requires every side to be shorter than the other two combined; 1, 1, 5 fails it. The tool checks first and explains, rather than producing an imaginary area.
Can sides be decimals or fractions?
Yes — all sides parse as exact rationals, and the radical simplification handles the fractions correctly. Runs locally.