About

Three known angles define a triangle's shape but not its size. An AAA (Angle-Angle-Angle) specification produces an infinite family of similar triangles sharing identical angle measures but differing in scale. This calculator validates that α + β + γ = 180°, classifies the triangle by angle type and symmetry, and computes side ratios via the Law of Sines. Without at least one known side length, no unique solution exists. Supplying an optional reference side a resolves the ambiguity and yields absolute dimensions, area, perimeter, inradius, and circumradius.

Errors in angle input propagate nonlinearly into derived quantities. A 1° mistake in an obtuse angle near 179° produces a degenerate triangle with near-zero area. This tool enforces strict validation: each angle must satisfy 0 < θ < 180°, and their sum must equal exactly 180° within floating-point tolerance. Pro tip: if working from a measured physical triangle, verify angle sum first to detect measurement drift before trusting derived values.

Formulas

The fundamental constraint for any Euclidean triangle is that its interior angles sum to 180° (π radians):

α + β + γ = 180°

The Law of Sines relates side lengths to the sines of their opposite angles, establishing fixed ratios for any AAA-specified triangle:

asin α = bsin β = csin γ = 2R

Given a reference side a, the remaining sides resolve to:

b = a ⋅ sin βsin α

c = a ⋅ sin γsin α

The area with two sides and included angle:

A = 12 ⋅ a ⋅ b ⋅ sin γ

The circumradius R and inradius r:

R = a2 ⋅ sin α

r = As

Where s = a + b + c2 is the semi-perimeter. Variable legend: α, β, γ = interior angles opposite sides a, b, c respectively. R = circumradius. r = inradius. A = area. s = semi-perimeter.

Reference Data

Triangle Type	Angle Condition	Side Condition	Example Angles	Properties
Equilateral	α = β = γ = 60°	a = b = c	60°, 60°, 60°	Maximum area for given perimeter
Isosceles Acute	Two angles equal, all < 90°	Two sides equal	70°, 70°, 40°	Line of symmetry exists
Isosceles Right	Two angles = 45°, one = 90°	Two legs equal	45°, 45°, 90°	Hypotenuse = leg × √2
Isosceles Obtuse	Two angles equal, one > 90°	Two sides equal	20°, 20°, 140°	Obtuse vertex opposite longest side
Scalene Acute	All angles different, all < 90°	All sides different	50°, 60°, 70°	Circumcenter inside triangle
Scalene Right	All different, one = 90°	All sides different	30°, 60°, 90°	Pythagorean theorem applies
Scalene Obtuse	All different, one > 90°	All sides different	25°, 35°, 120°	Circumcenter outside triangle
30-60-90 Special	30°, 60°, 90°	1 : √3 : 2	30°, 60°, 90°	Half of an equilateral triangle
45-45-90 Special	45°, 45°, 90°	1 : 1 : √2	45°, 45°, 90°	Diagonal of a square
Golden Gnomon	36°, 36°, 108°	Ratio involves φ	36°, 36°, 108°	Appears in regular pentagons
Golden Triangle	72°, 72°, 36°	Ratio involves φ	72°, 72°, 36°	Self-similar bisection
Near-Degenerate	One angle → 0° or 180°	One side → 0 or ∞	1°, 1°, 178°	Area approaches zero
Degenerate	One angle = 180°	Collinear points	0°, 0°, 180°	Not a valid triangle

Frequently Asked Questions

Three angles define a triangle's shape (its similarity class) but not its scale. Any triangle with angles 40°, 60°, 80° is similar to every other triangle with those same angles, regardless of whether its shortest side is 1 cm or 1 km. You need at least one side length to fix the scale factor and obtain a unique (congruent) solution.

When one angle approaches 0° or 180°, the sine of that angle approaches 0, causing side ratios to diverge toward infinity. For angles below approximately 0.5°, IEEE 754 double-precision floating point introduces relative errors exceeding 0.01% in the sine computation. This calculator rejects angles below 0.01° to avoid numerically meaningless output.

The Law of Sines states a / sin α = b / sin β = c / sin γ. Without a known side, the common ratio is unknown, but the relative proportions sin α : sin β : sin γ are fully determined. This calculator normalizes these to the smallest value equaling 1, giving a clean ratio like 1 : 1.414 : 1.732.

No. This tool assumes a flat Euclidean plane where interior angles sum to exactly 180°. On a sphere (elliptic geometry), the angle sum exceeds 180° by an amount proportional to the triangle's area (the spherical excess). On a hyperbolic plane, the sum is strictly less than 180°. For geodetic surveying on Earth's surface, triangles spanning more than roughly 100 km per side require spherical corrections.

By angle: if any angle equals exactly 90°, the triangle is right. If any angle exceeds 90°, it is obtuse. Otherwise all angles are below 90° and it is acute. By symmetry: if all three angles are equal (60° each), it is equilateral. If exactly two are equal, it is isosceles. If all three differ, it is scalene. These two classifications combine (e.g., isosceles right, scalene obtuse).

The circumradius R = a / (2 ⋅ sin α). Since sin α is known from angles alone, R scales linearly with the reference side a. For an equilateral triangle with side a, R = a / √3. The circumradius equals the radius of the circle passing through all three vertices.