12. Heron's Formula Triangle Area Calculator — Theory
Heron’s formula lets you compute the area of any triangle using only the three side lengths
\(a\), \(b\), and \(c\) — no angles or height required.
1) Triangle Inequality (Validity Check)
Before using Heron’s formula, the side lengths must form a valid triangle. The key condition is:
\[
a+b>c,\quad a+c>b,\quad b+c>a
\]
If the smallest two sides add up to exactly the largest side, the triangle is degenerate and its area is \(0\).
2) Semi-perimeter
Define the semi-perimeter \(s\) as half the perimeter:
\[
s=\frac{a+b+c}{2}
\]
3) Heron’s Formula
The area \(A\) is:
\[
A=\sqrt{s(s-a)(s-b)(s-c)}
\]
Worked Example
Sides \(a=5\), \(b=6\), \(c=7\).
\[
s=\frac{5+6+7}{2}=\frac{18}{2}=9
\]
\[
A=\sqrt{9(9-5)(9-6)(9-7)}=\sqrt{9\cdot 4\cdot 3\cdot 2}=\sqrt{216}\approx 14.70
\]
So the area is about \(14.7\) square units.
Notes and Tips
-
Heron’s formula is numerically stable for typical values, but extreme side ratios can cause rounding issues.
-
If you also want the triangle type, you can compare squares of side lengths (after sorting):
- Right: \(x^2+y^2=z^2\)
- Acute: \(x^2+y^2 \gt z^2\)
- Obtuse: \(x^2+y^2 \lt z^2\)
-
The calculator diagram constructs a triangle schematically from \(a,b,c\) (it is not “scaled from angles,” only from side lengths).
