The area of a triangle can be found in different ways depending on what information is known.
Two very useful formulas are Heron's formula and the trigonometric area formula.
1. Heron's formula
Heron's formula is used when all three side lengths are known.
First compute the semiperimeter:
\[
s=\frac{a+b+c}{2}
\]
Then compute the area:
\[
K=\sqrt{s(s-a)(s-b)(s-c)}
\]
Here, \(K\) represents the triangle area.
2. Trigonometric area formula
If two sides and the included angle are known, use:
\[
K=\frac12ab\sin C
\]
This works because the height relative to side \(a\) or \(b\) can be written using the sine of the included angle.
Similar forms are:
\[
K=\frac12bc\sin A
\]
\[
K=\frac12ac\sin B
\]
\[
K=\frac12ab\sin C
\]
3. Base-height formula
The familiar area formula is:
\[
K=\frac12(\text{base})(\text{height})
\]
Heron's formula and the trigonometric formula are both connected to this base-height idea.
4. Example with Heron's formula
Suppose \(a=7\), \(b=8\), and \(c=9\). Then:
\[
s=\frac{7+8+9}{2}=12
\]
\[
K=\sqrt{12(12-7)(12-8)(12-9)}
\]
\[
K=\sqrt{12\cdot5\cdot4\cdot3}
\]
\[
K=\sqrt{720}
\]
5. Example with the trigonometric formula
Suppose \(a=5\), \(b=7\), and \(C=60^\circ\). Since \(C\) is the included angle between sides
\(a\) and \(b\), use:
\[
K=\frac12ab\sin C
\]
\[
K=\frac12(5)(7)\sin60^\circ
\]
6. Formula summary
| Known information |
Formula |
Best method |
| Three sides \(a,b,c\) |
\(K=\sqrt{s(s-a)(s-b)(s-c)}\) |
Heron's formula |
| Two sides \(a,b\) and included angle \(C\) |
\(K=\dfrac12ab\sin C\) |
Trigonometric formula |
| Base and height |
\(K=\dfrac12bh\) |
Base-height formula |
7. Triangle validity
If three sides are used, they must satisfy the triangle inequality:
\[
a+b>c,\qquad a+c>b,\qquad b+c>a
\]
If these inequalities fail, the sides cannot form a real triangle.
8. Common mistakes
- In \(K=\frac12ab\sin C\), angle \(C\) must be the included angle between sides \(a\) and \(b\).
- Do not use Heron's formula unless all three sides are known.
- Check that three side lengths can actually form a triangle.
- Keep angle units consistent: degrees with degrees, radians with radians.
- Area is always positive; a negative or zero result means the input is invalid or degenerate.
