How this calculator works
The perpendicular height contributed by side b is b sin(C), so multiplying by a and halving gives the area. Reversing for a side requires division by the sine. Reversing for C requires an inverse sine, but an acute angle and its supplementary obtuse angle share the same sine. The table lists both triangles, including their different third sides. The maximum area occurs at a right angle.
Formula
K = ab sin(C)/2; a = 2K/[b sin(C)]; sin(C) = 2K/(ab); C₂ = π - C₁; K_max = ab/2.
Worked example
Sides 4 m and 6 m with C = 30° give 6 m². Reversing that area yields C = 30° or 150°. With these same sides the maximum area is 12 m² at C = 90°.
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Frequently asked questions
Can I use an angle opposite one of the known sides?
No. This formula needs the included angle where the two known sides meet.
Why are there two angles for one area?
Sine has the same value at C and 180° - C. Both angles enclose the same area between the given sides, with different third sides.
When is there only one reverse solution?
When area equals ab/2, the two candidates coincide at 90°. A larger area is impossible.