The triangle inequality
Not every set of three numbers makes a triangle. Any two sides must add up to more than the third — otherwise the two shorter sides cannot reach across the longest one. Sides of 1, 2 and 5 simply do not close, and 1, 2 and 3 collapse into a straight line rather than a shape.
The diagram is what makes this obvious. When the sides are valid you see the triangle; when they are not you get told why, rather than a plausible-looking number for a shape that cannot exist.
Heron's formula and the law of cosines
The area comes from Heron's formula, which needs only the three sides: take the semi-perimeter s, then √(s(s−a)(s−b)(s−c)). No height required, which is exactly why it is useful — measuring a height means first finding where the perpendicular lands.
The angles come from the law of cosines, cos A = (b² + c² − a²) ÷ 2bc. When one angle works out to exactly 90° the diagram switches from an arc to the right-angle square, and the Pythagorean theorem is just this formula with that cosine equal to zero.
Inradius and circumradius
The inradius is the radius of the largest circle that fits inside, area ÷ s. The circumradius is the radius of the circle through all three corners, abc ÷ 4×area. For a right triangle the circumradius is always exactly half the hypotenuse — the right angle sits on the circle's diameter.
Frequently asked questions
Can I enter angles instead of sides?
Not on this page — it solves from three sides (the SSS case). Three angles alone never determine a triangle's size, only its shape, since any scaled copy has identical angles.
Is anything uploaded?
No. The diagram is drawn from your measurements, in the browser.
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