How this calculator works
With sides, the cosine rule recovers the angles; with two angles, their sum supplies the third. AAA checks the supplied total and reports no physical lengths or area. The margin is 90 degrees minus the largest angle: positive means acute, zero means right and negative means obtuse. Angles within 10^-12 radians of 90 degrees are treated as right for this check.
Formula
cos(A) = (b² + c² - a²)/(2bc); A + B + C = π; acute margin = π/2 - max(A,B,C).
Worked example
Sides 5 m, 5 m and 6 m enclose 12 m². The largest angle is about 73.739795°, leaving a 16.260205° margin. Sides 3, 4 and 5 instead have zero margin and are not acute.
Advertisement
Frequently asked questions
Can a triangle have only two acute angles?
Every non-degenerate right or obtuse triangle has two acute angles, but an acute triangle requires all three to be acute.
Does AAA give area?
No. Angles alone leave the scale unknown. Use SSS, SAS or ASA to set the size.
What does a negative margin mean?
At least one angle exceeds 90 degrees. Its magnitude is the excess of the largest angle above a right angle.