How this instrument works
Given a triangle's three side lengths, with c as the longest, the sign of a² + b² − c² reveals the triangle's category without ever computing an actual angle. A result of exactly 0 means a right triangle (the ordinary Pythagorean theorem holding exactly). A NEGATIVE result means an obtuse triangle, with the angle opposite the longest side exceeding 90°. A POSITIVE result means an acute triangle, every angle strictly under 90°.
This works because a² + b² − c² is proportional to the cosine of the angle opposite side c, by the Law of Cosines rearranged: cos(C) = (a²+b²−c²) ⁄ 2ab. Since cosine is positive for angles under 90°, exactly zero at 90°, and negative for angles over 90°, the sign of the numerator alone — without ever dividing through or taking an arccosine — already answers the classification question.
Only the angle opposite the LONGEST side needs checking, since it's automatically the triangle's largest angle (a longer side always sits opposite a larger angle) — if that one angle isn't obtuse, none of the triangle's other, smaller angles could be obtuse either, so a single test on the longest side settles the whole triangle's classification.
- Enter the two shorter side lengths into the Side a and Side b fields.
- Enter the longest side length into the Side c field.
- Read the test value: negative means obtuse, exactly zero means right, and positive means acute.
Worked example — sides 4, 5, and 7
Sides 4, 5, and 7 (with 7 the longest) give a test value of 16+25−49 = −8, a NEGATIVE result — this is an OBTUSE triangle, meaning the angle opposite the 7-unit side is greater than 90°.
The classic 3-4-5 triangle gives a test value of exactly 0 — a RIGHT triangle, the precise boundary between acute and obtuse. Sides 4, 5, and 6 instead give a test value of +5, a POSITIVE result — an ACUTE triangle, every angle strictly less than 90°.
Questions
How do you tell if a triangle is obtuse just from its sides?
Compute a² + b² − c², with c as the longest side. A negative result means the triangle is obtuse — the angle opposite the longest side exceeds 90° — with no need to compute the actual angle to know it.
Why does this formula work without computing an actual angle?
Because a²+b²−c² is directly proportional to the cosine of the angle opposite c, by the Law of Cosines rearranged. Since cosine's sign alone (positive, zero, or negative) already tells whether an angle is under, exactly at, or over 90°, checking the sign of this simpler expression is enough.
Why only check the angle opposite the longest side?
Because a longer side always sits opposite a larger angle — the angle opposite the longest side is automatically the triangle's biggest one. If that particular angle isn't obtuse, none of the other two (necessarily smaller) angles could be obtuse either.
What does a test value of exactly zero mean?
The triangle is a RIGHT triangle — the boundary case between acute and obtuse, where the angle opposite the longest side is exactly 90°, and a²+b²−c² becomes exactly zero, the ordinary Pythagorean theorem holding precisely.
Can a triangle have more than one obtuse angle?
No — a triangle can have at most one angle of 90° or more, since the three angles must sum to exactly 180° and two angles of 90°+ would already exceed that total on their own. This is exactly why checking only the largest angle is sufficient.