How this instrument works
The test rests on a single rearrangement of the Law of Cosines. For any triangle, c² = a² + b² − 2ab·cos(C), where C is the angle opposite whichever side is labeled c. Since a and b are positive lengths, the sign of cos(C) — and therefore whether C is sharp, square, or wide — matches the sign of a² + b² − c². This calculator's formula, a² + b² + c² − 2·max(a², b², c²), packages that same comparison so it works regardless of which of the three inputs happens to be longest: subtracting twice the largest square cancels it out of the sum, leaving the sum of the two shorter squares minus the longest one — the same quantity the Law of Cosines checks, with the biggest side picked out automatically instead of by hand.
This converse of the Pythagorean theorem is old. Euclid proved both directions in the Elements, Book II: Proposition 12 shows that if the square on the longest side exceeds the sum of the other two squares, the opposite angle is obtuse, and Proposition 13 proves the acute case. Euclid's argument drops a perpendicular from one vertex and measures how far its foot lands outside or inside the opposite side — the same geometric fact the Law of Cosines later packaged into a single algebraic line.
A degenerate limit sits at the far end of the scale: as the longest side grows toward the sum of the other two, the triangle flattens toward a straight line and the classification value plunges further negative, never quite reaching zero area while a genuine triangle survives. The mistake this test guards against is comparing the wrong pair of lengths — squaring and subtracting before checking which one is actually longest can flip an obtuse triangle into a false acute reading; the max() in the formula removes that judgment call entirely.
- Enter the triangle's three side lengths into Side a, Side b, and Side c, in any order — the calculator works out which one is longest for you.
- Read Classification value: a positive number means acute, zero means right, and a negative number means obtuse.
- Look at the size of the number, not just its sign — a value near zero sits close to a right angle, while a large negative value marks a noticeably wide, flattened triangle.
- If the three lengths can't actually close into a triangle, a warning appears instead of a result; adjust Side a, Side b, or Side c until it clears.
Worked example — the 5-5-5 equilateral triangle
Take an equilateral triangle with all three sides equal to 5. Squaring and summing gives a² + b² + c² = 25 + 25 + 25 = 75, and the longest side — tied three ways at 5 — squares to 25, so 2·max(a², b², c²) = 2 × 25 = 50. The classification value comes out to 75 − 50 = 25, comfortably positive, so the sheet reports an acute triangle.
That result generalizes well past this one triangle: every equilateral triangle, whatever its side length s, produces a classification value of exactly s squared, always positive. It is also why no equilateral triangle can ever be right or obtuse — with all three angles pinned to 60° by symmetry, none of them ever reaches the 90° threshold this test is built to detect.
Questions
How does comparing squares tell you whether an angle is obtuse or acute?
It relies on the Law of Cosines, c² = a² + b² − 2ab·cos(C). Because a and b are always positive, the sign of a² + b² − c² matches the sign of cos(C) exactly: positive means C is under 90°, zero means C is exactly 90°, and negative means C is over 90°. This calculator runs that comparison automatically against whichever side turns out to be longest.
Why does this test only need the three side lengths, not any angles?
Three side lengths already pin down a triangle's shape completely, a fact usually called the SSS congruence result, so every angle is implicitly fixed the moment the three sides are chosen. The Law of Cosines makes that implicit angle information explicit without a protractor, which is exactly why side lengths alone are enough here.
What is the most common mistake when doing this test by hand?
Subtracting the wrong square. The test only works if you double the square of the longest length and subtract that from the sum of all three squares; comparing a shorter one instead can flip an obtuse triangle into a false acute reading. This calculator's max() term removes that step, finding the longest length automatically every time.
Does a positive classification value also confirm the three sides form a real triangle?
No, that is a separate question known as the triangle inequality, which just checks that each length is shorter than the sum of the other two. This sheet checks that first and flags any set of lengths that cannot close into a triangle at all; the acute-right-obtuse test only runs once a genuine triangle is confirmed.
Is an equilateral triangle always acute?
Yes, without exception. A 5-5-5 triangle gives a classification value of 25, which is 5 squared, and the same pattern holds for any equilateral triangle: with every angle forced to exactly 60° by symmetry, none can ever reach the 90° boundary that separates acute from right or obtuse.
What happens to the value as a triangle gets close to flat?
It grows more negative the closer a triangle gets to degenerate. A 2-2-3.9 triangle, just short of the 2 + 2 = 4 limit where the shape collapses to a line, gives a classification value of about −7.21 — clearly obtuse and visibly thin, well before the sides stop being able to close at all.