How this calculator works
The semiperimeter is half the boundary length. Multiplying it by its three differences from the sides gives area squared. The factor table shows those lengths in metres so you can follow the substitution; the area readout has its own unit selector. Strict triangle inequalities keep all four factors positive. In reverse area mode, K = ab sin C/2 gives an acute and an obtuse included angle when both are possible. The tool evaluates a rearranged Heron product internally to reduce cancellation for thin triangles.
Formula
s=(a+b+c)/2; K=√[s(s-a)(s-b)(s-c)]. From perimeter: c=P-a-b. From area and two sides: sin C=2K/(ab), followed by c²=a²+b²-2ab cos C.
Worked example
For sides 13 m, 14 m and 15 m, s=21 m and K=√(21×8×7×6)=84 m². For 3 m, 4 m and 5 m, the factors are 6, 3, 2 and 1 m, giving 6 m².
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Frequently asked questions
Why use Heron instead of base times height?
It uses side lengths only, which is helpful when the perpendicular height is difficult to measure.
Can area and two sides identify two triangles?
Yes. Equal sine values at supplementary included angles can give the same area with different third sides. Both solutions appear in the table.
What does a zero factor mean?
The triangle is flat. A genuine triangle requires every factor in Heron’s product to be positive.