How this instrument works
AAA (angle-angle-angle) refers to knowing all three of a triangle's interior figures at once, and this calculator answers the most basic question that situation raises: do these three values actually describe a real triangle at all? The test is simple — add all three together and subtract 180°. A result of exactly 0 means yes, a genuinely valid triangle; anything else means the three values are geometrically impossible together.
This is a different question from the already-live Triangle Sum Theorem page elsewhere on this site, which starts from just TWO known corners and solves for the missing third. Here, all three are already given, and the only thing left to determine is whether they're mutually consistent — a validation check rather than a completion.
It's worth knowing what AAA does NOT determine: three valid measures fix a triangle's SHAPE completely (every triangle sharing those same three values is similar to every other one), but they say nothing at all about its SIZE — a tiny triangle and an enormous one can share identical corner measures. This is the key reason AAA, unlike SSS or SAS, is not a congruence condition; it's only a similarity condition.
- Enter the first angle into the Angle A field.
- Enter the second angle into the Angle B field.
- Enter the third angle into the Angle C field.
- Read the test value: exactly 0° means these three angles form a valid triangle; any other value means they don't.
Worked example — angles of 60°, 60°, and 60°
Three measures of 60°, 60°, and 60° give a test value of exactly 0° — a valid triangle, specifically an equilateral one, since all three sum to precisely 180° as every triangle's interior corners must.
Compare 90°, 45°, and 50°: these sum to 185°, giving a test value of +5°, clearly nonzero — an INVALID combination. No real triangle can have interior figures totaling anything other than exactly 180°, so this particular trio simply cannot describe a genuine triangle.
Questions
What does AAA mean for a triangle?
AAA stands for angle-angle-angle — knowing all three of a triangle's interior corners. This calculator checks whether three given values are even mutually consistent, since every valid triangle's three corners must sum to exactly 180°.
Why must three angles sum to exactly 180°?
This is the Triangle Sum Theorem, a fundamental fact of flat (Euclidean) geometry: draw a line through one vertex parallel to the opposite side, and the resulting alternate-interior figures recreate the other two corners laid alongside the original along that straight line, which is itself always 180°.
Does AAA determine a triangle's size?
No — AAA only fixes a triangle's SHAPE, not its size. Any two triangles sharing the same three corner values are similar (proportionally identical), but one could be tiny and the other enormous, since those values alone say nothing about actual side lengths.
How is AAA different from the Triangle Sum Theorem calculator?
The Triangle Sum Theorem page starts from two known angles and solves for the missing third. This page instead starts with all three angles already given and simply validates whether they're consistent — a check rather than a completion.
Is AAA a valid triangle congruence condition?
No — AAA is only a SIMILARITY condition, not a congruence one, precisely because it fixes shape but not size. SSS, SAS, and ASA, by contrast, are all valid congruence conditions, since each one pins down both the shape and the exact size of a triangle.