Heron's Formula Calculator

Calculate the area of a triangle from three side lengths using Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter. No height needed.

📐 Geometry📐 A = √(s(s-a)(s-b)(s-c))🔢 Math
Side a
Side b
Side c
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Heron's Formula Calculator—A = √(s(s-a)(s-b)(s-c))square units

Step-by-Step Examples

Example 1
General Triangle

a=5, b=6, c=7.

  • s=9
  • A=√(9×4×3×2)=√216≈14.70
✓ A ≈ 14.70 sq units
Example 2
Right Triangle Check

a=3, b=4, c=5.

  • s=6
  • A=√(6×3×2×1)=√36=6
  • Verify: ½×3×4=6 ✓
✓ A = 6 sq units
Example 3
Equilateral

a=b=c=6.

  • s=9
  • A=√(9×3×3×3)=9√3≈15.59
  • Matches A=√3/4×36≈15.59 ✓
✓ A ≈ 15.59 sq units

Real-World Applications

📐
Surveying
Calculate land area from three measured boundary lengths.
🏗️
Engineering
Triangular cross-sections of beams and structural members.
🎮
Computer Graphics
Triangle rasterization uses Heron's formula for area checks.
📊
Trigonometry
Paired with law of cosines to solve oblique triangles.

Common Mistakes to Avoid

⚠️
Triangle inequality must hold

a+b>c, a+c>b, b+c>a. If any fails, no triangle exists. The formula will give a negative radicand.

⚠️
s is semi-perimeter

s=(a+b+c)/2, not the full perimeter. Using perimeter instead of semi-perimeter is a common error.

⚠️
Numerical precision for nearly degenerate triangles

If the triangle is very flat (near-zero area), floating-point errors can cause s−c≈0. Use Kahan's robust formula for better precision.

Frequently Asked Questions

Who discovered Heron's formula? ▾
Heron of Alexandria (around 60 AD), though it may have been known to Archimedes earlier.
When to use Heron's vs ½bh? ▾
Use Heron's when you know all three sides but not the height. If you know a base and corresponding height, use A=½bh.
How is it derived? ▾
Using the cosine rule to find an angle, then computing area from ½ab×sin(C). Heron's formula is the result after simplification.
What is the Cayley-Menger determinant? ▾
Generalizes Heron's formula to n dimensions. For tetrahedra in 3D, it computes volume from six edge lengths.
Can Heron's formula work in 3D? ▾
Heron's formula gives the area of any triangle, whether in 2D or 3D space (just provide the three side lengths).

Related Math Calculators

📐 Trapezoid📐 Pythagorean Theorem📐 Law of Cosines📐 Coordinate Geometry📐 Triangle

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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