AAA Triangle Calculator (All Three Angles)
Enter all three angles of a triangle; optionally supply one side length to scale and fully solve the triangle using the law of sines.
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All sides computed from the reference side via law of sines
How does this calculator work?
Three angles A, B, C are valid only if they are all positive and A + B + C = 180°. Supply side a to scale: b = a sin B / sin A, c = a sin C / sin A. Area follows from Heron's formula; circumradius = a / (2 sin A). Without a side, only angle properties are available — all similar triangles share the same three angles.
Formula
How this is calculated
Knowing only three angles (the AAA case) fixes the shape of the triangle but not its size — any triangle with those angles is geometrically similar to every other. The key constraint is that the angles must sum to exactly 180°. Without a side length you can confirm the triangle is valid and find all angle relationships, but you cannot determine actual side lengths, area, or perimeter.
This calculator verifies the angle sum and then, if you supply side a (the side opposite angle A), applies the law of sines to scale the whole triangle: b = a × sin B / sin A and c = a × sin C / sin A. The common ratio a / sin A equals the circumdiameter (2R). Heron's formula gives the area — s = (a+b+c)/2, area = √(s(s−a)(s−b)(s−c)) — while the inradius is area / s and the circumradius is a / (2 sin A).
All angles must be strictly positive and sum to 180°. Angles of exactly 0° or 180° would make the triangle degenerate (a straight line). Because AAA specifies only ratios, the visualisation shows the triangle shape with either angle labels or computed side lengths, depending on whether a reference side was provided.
Frequently asked questions
No — three angles determine only the shape (all similar triangles share the same angles) but never the size. You need at least one actual side as a reference. Enter it in the "Side a" field and the law of sines scales the other two sides.
Check that all three angles are strictly positive (greater than 0°). An angle of exactly 0° or 180° makes the triangle degenerate — a straight line rather than a closed polygon — so those cases are rejected.
AAA gives only the shape; you still need a side to determine size. AAS (two angles and a non-included side) provides a real length directly, so the law of sines fully determines the triangle without any ambiguity.
Also known as
TG we-Calculate Editorial Team. (2026). AAA Triangle Calculator (All Three Angles) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/aaa-triangle-calculator
TG we-Calculate Editorial Team. "AAA Triangle Calculator (All Three Angles)." TG we-Calculate. 2026. https://we-calculate.com/calculator/aaa-triangle-calculator.
TG we-Calculate Editorial Team, "AAA Triangle Calculator (All Three Angles)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/aaa-triangle-calculator
@misc{wecalculate_aaa_triangle_calculator, title = {AAA Triangle Calculator (All Three Angles)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/aaa-triangle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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