Heron's Formula Triangle Area Calculator
Heron's Formula computes the exact area A of any triangle directly from its three side lengths a, b, and c using the semi-perimeter s = (a + b + c) / 2 without requiring angle or altitude measurements.
Enclosed planar area of the triangle
First side of the triangle
Second side of the triangle
Third side of the triangle
Step-by-Step Calculation
Heron's Formula computes the exact area A of any triangle directly from its three side lengths a, b, and c using the semi-perimeter s = (a + b + c) / 2 without requiring angle or altitude measurements.
Real-World Reference Values
Famous integer-side integer-area triangle (altitude = 12 on base 14)
Radius of inscribed circle tangent to all three sides
Any two sides must sum to strictly more than the third
20-Row Comparison Table
Triangle Area (A) · Side Length a (a)
| # | Side Length a (m) | Triangle Area (m²) | Δ % |
|---|---|---|---|
| 01 | 13 mCurrent | 84 m² | 0.00% |
| 02 | 36.9624 m | 0 m² | -100.00% |
| 03 | 73.8249 m | 0 m² | -100.00% |
| 04 | 110.6873 m | 0 m² | -100.00% |
| 05 | 147.5497 m | 0 m² | -100.00% |
| 06 | 184.4122 m | 0 m² | -100.00% |
| 07 | 221.2746 m | 0 m² | -100.00% |
| 08 | 258.137 m | 0 m² | -100.00% |
| 09 | 294.9995 m | 0 m² | -100.00% |
| 10 | 331.8619 m | 0 m² | -100.00% |
| 11 | 368.7243 m | 0 m² | -100.00% |
| 12 | 405.5868 m | 0 m² | -100.00% |
| 13 | 442.4492 m | 0 m² | -100.00% |
| 14 | 479.3116 m | 0 m² | -100.00% |
| 15 | 516.1741 m | 0 m² | -100.00% |
| 16 | 553.0365 m | 0 m² | -100.00% |
| 17 | 589.8989 m | 0 m² | -100.00% |
| 18 | 626.7614 m | 0 m² | -100.00% |
| 19 | 663.6238 m | 0 m² | -100.00% |
| 20 | 700.4863 m | 0 m² | -100.00% |
Frequently Asked Questions
How does the calculator invert Heron's formula to solve for a side?+
Given area A and two sides b and c, sin(α) = 2A/(bc), which yields cos(α) = √(1 - sin²α) and substitutes directly into the Law of Cosines a = √(b² + c² - 2bc cos α).
What is a Heronian triangle?+
A Heronian triangle is a triangle whose side lengths a, b, c and area A are all rational integers (such as 3-4-5 with area 6, or 13-14-15 with area 84).
How do I find the altitude (height) to base b from Heron's area?+
Since A = ½ × b × h_b, the perpendicular altitude to side b is simply h_b = 2A / b.