How this instrument works
Every triangle carries two independent classifications at once: one by its angles (acute, right, or obtuse) and one by its sides (scalene, isosceles, or equilateral). Most calculators settle only one of those questions per visit. Feed the three lengths a, b, and c into this sheet and it hands back both answers from the same set of measurements, so a shape that turns out obtuse and scalene, or acute and equilateral, gets named completely in a single pass rather than two separate lookups.
The angle classification comes from a rearranged Law of Cosines: angle test = a² + b² − c², with c held as the longest of the three lengths. A negative result marks an obtuse triangle, since the corner opposite that longest length has widened past ninety degrees. Exactly zero marks a right triangle, the ordinary Pythagorean relation holding on the nose. A positive result marks an acute triangle, every corner still under ninety degrees.
The second figure, side test = (a−b)(b−c)(c−a), answers a different question entirely: are any two of the three lengths identical? Each factor is a gap between one pair of lengths, so a repeated length zeroes out its own factor and drags the whole product to zero. A nonzero result therefore certifies a scalene shape, with no pair matching; a result of exactly zero flags an isosceles pair or, if every factor vanishes, an equilateral trio.
This calculator's whole reason for existing is combination. The obtuse-triangle sheet already live on the site runs the angle test alone, and the scalene-triangle sheet runs the side test alongside Heron's-formula area, but neither one reports the other family at all. Here, both tests run from the same three inputs at once, which suits anyone who wants a complete label — say, 'obtuse scalene' — rather than two separate visits to two separate pages just to describe one figure fully.
- Enter the two shorter lengths into Side a and Side b, in any consistent unit.
- Enter the longest of the three lengths into Side c (the longest).
- Read Angle test: negative means obtuse, zero means right, positive means acute.
- Read Side test: zero means at least two sides match; nonzero confirms a scalene shape.
- Combine both readings for a full label, such as acute scalene or obtuse isosceles.
Worked example — a 3-4-5 triangle, fully labeled
Sides 3, 4, and 5 give an angle test of 3²+4²−5² = 9+16−25 = 0, exactly zero, so this is a RIGHT triangle. The same three lengths give a side test of (3−4)(4−5)(5−3) = (−1)(−1)(2) = 2, clearly nonzero, so this is also a SCALENE triangle — the full label is 'right scalene,' read off both tests in one pass rather than two separate calculators.
Change the lengths to 4, 5, and 7 and both answers shift: the angle test becomes 4²+5²−7² = 16+25−49 = −8, a negative result marking an OBTUSE triangle, while the side test becomes (4−5)(5−7)(7−4) = (−1)(−2)(3) = 6, still nonzero, so this second shape is 'obtuse scalene.' A third case, sides 5, 5, and 5, gives an angle test of 25+25−25 = 25 (ACUTE, since it's positive) and a side test of exactly 0, since every factor in (5−5)(5−5)(5−5) vanishes — an 'acute equilateral' figure, the plainest possible full label this pair of tests can return.
Questions
What does a negative angle test actually mean?
It means the triangle is obtuse — the corner opposite the longest length, c, has grown past ninety degrees. Sides 4, 5, and 7 give an angle test of 16+25−49 = −8, negative, confirming an obtuse shape without ever computing the angle itself in degrees.
What does a side test of zero tell me?
It tells you at least two of the three lengths match, so the shape is isosceles rather than scalene — or, if all three lengths agree, equilateral. Sides 5, 5, and 5 give (5−5)(5−5)(5−5) = 0, since every one of the three factors already sits at zero.
How is this different from the obtuse-triangle page on the site?
That page runs the angle test alone and stops at acute, right, or obtuse. This page runs the identical angle test but pairs it with a second, independent side test, so the result also names the figure as scalene, isosceles, or equilateral — a full classification from one visit instead of a partial one.
How is this different from the scalene-triangle page on the site?
That page pairs the side test with Heron's-formula area, answering whether the shape is scalene and how big it is. This page skips area entirely and pairs the side test with the angle test instead, answering whether the shape is scalene and whether it is acute, right, or obtuse.
Can a triangle be both right and isosceles at once?
Yes — a right isosceles triangle has two legs of equal length meeting at a ninety-degree corner. Lengths of 5, 5, and roughly 7.07 (5√2) return an angle test of exactly 0 and a side test of exactly 0 too, since two of the three lengths match while the widest corner still lands at ninety degrees.
Does the order I enter the three lengths in matter?
The side test's sign can flip depending on which length is labeled a, b, or c, but whether it lands on zero or off zero never changes. The angle test needs c to hold the longest of the three lengths, or the classification it returns will not match the actual figure being described.