🔻 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.