How this calculator works
The perpendicular legs act as base and height. Side-and-angle modes reconstruct the legs using sine, cosine or tangent. Area-and-leg modes rearrange K = ab/2. With only area and hypotenuse, interchanging the two legs gives a second labelled solution; the shorter leg is reported as a. The largest possible area for a fixed hypotenuse occurs when the legs are equal.
Formula
K = ab/2; c² = a²+b²; a = c sin A; b = c cos A; K ≤ c²/4. With K and c: a+b = √(c²+4K), |a-b| = √(c²-4K).
Worked example
Legs 3 m and 4 m give area 6 m² and hypotenuse 5 m. An area of 30 m² with leg a = 5 m gives b = 12 m and c = 13 m.
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Frequently asked questions
Which angle do I enter?
A is the acute angle opposite leg a. If you know the other acute angle, subtract it from 90° first.
Can area and hypotenuse be inconsistent?
Yes. Area greater than c²/4 is impossible. At the limit both legs equal c/√2.
Why do I see two orientations?
Area and hypotenuse do not identify which leg is opposite A. The same physical shape has two assignments unless the legs are equal.