About

The AAS (Angle-Angle-Side) configuration defines a triangle by two known angles and one side that is not between them. This is a unique-solution case: given angles A and B with side a opposite A, the third angle C = 180° − A − B follows directly. All remaining sides are computed via the Law of Sines. Unlike the ambiguous SSA case, AAS always produces exactly one valid triangle provided A + B < 180°. Errors in angle measurement propagate non-linearly through the sine function. A 1° error near 90° barely affects results, but the same error near 0° or 180° causes catastrophic distortion in computed side lengths. This tool applies IEEE 754 double-precision arithmetic and reports all derived properties including circumradius, inradius, altitudes, and medians.

Formulas

The AAS solver derives all unknowns from two angles and the side opposite the first angle. The third angle follows from the Euclidean angle sum property. All remaining sides are computed via the Law of Sines.

Step 1 - Third Angle:

C = 180° − A − B

Step 2 - Law of Sines:

asin A = bsin B = csin C

b = a ⋅ sin Bsin A

c = a ⋅ sin Csin A

Step 3 - Area:

Area = 12 ⋅ b ⋅ c ⋅ sin A

Step 4 - Perimeter:

P = a + b + c

Step 5 - Circumradius:

R = a2 ⋅ sin A

Step 6 - Inradius:

r = Areas , where s = P2

Step 7 - Altitudes:

h_a = 2 ⋅ Areaa , h_b = 2 ⋅ Areab , h_c = 2 ⋅ Areac

Step 8 - Medians:

m_a = 12 √2b² + 2c² − a²

Where A, B, C are the interior angles in degrees. a, b, c are the sides opposite their respective angles. R is the circumradius. r is the inradius. s is the semi-perimeter. h_x is the altitude to side x. m_x is the median to side x.

Reference Data

Triangle Type	Angle Condition	Properties	Example Angles
Acute	All angles < 90°	Circumcenter inside triangle	60°, 70°, 50°
Right	One angle = 90°	Hypotenuse is diameter of circumcircle	90°, 45°, 45°
Obtuse	One angle > 90°	Circumcenter outside triangle	120°, 35°, 25°
Equilateral	All angles = 60°	All sides equal, maximum area for perimeter	60°, 60°, 60°
Isosceles	Two angles equal	Two sides equal, axis of symmetry	70°, 70°, 40°
Scalene	All angles different	No sides equal, no symmetry	50°, 60°, 70°
Degenerate	A + B = 180°	Collinear points, zero area	90°, 90°, 0°
Key Constants & Formulas
Angle Sum	A + B + C = 180° (π rad)
Law of Sines ratio	asin A = 2R
Area (two sides)	12 b c sin A
Inradius	r = Areas where s = P2
Circumradius	R = a2 sin A
Altitude to side a	h_a = 2 ⋅ Areaa
Median to side a	m_a = 12 √2b² + 2c² − a²
Degrees to Radians	1° = π180 rad ≈ 0.01745 rad
sin values	sin 30° = 0.5	sin 45° = 0.7071	sin 60° = 0.8660
sin values	sin 90° = 1.0	sin 120° = 0.8660	sin 150° = 0.5

Frequently Asked Questions

The triangle becomes degenerate or impossible. If A + B = 180°, the third angle C is zero, producing a line segment with zero area. If A + B > 180°, no Euclidean triangle exists. The calculator rejects both cases and reports a validation error.

In ASA, the known side lies between the two known angles. In AAS, the known side is opposite one of the known angles, not between them. Mathematically both produce a unique triangle because knowing two angles immediately gives the third. The distinction matters for labeling: in AAS you apply the Law of Sines directly using the side-angle pair, while in ASA you first identify which side is opposite which angle. This calculator handles AAS natively - side a is opposite angle A.

The Law of Sines divides by sin A. As A approaches 0°, sin A approaches 0, making b and c grow toward infinity. Physically, a triangle with a near-zero angle is extremely elongated. Measurement precision matters most in this regime: a 0.1° error at A = 2° changes the result by roughly 5%, while the same error at A = 60° changes it by about 0.1%.

No. This tool assumes Euclidean (flat plane) geometry where the angle sum is exactly 180°. On a sphere, the angle sum exceeds 180° by the spherical excess E = A + B + C − 180°, and the Law of Sines takes a different form involving the sphere's radius. For geodetic or astronomical triangles, use a spherical trigonometry tool.

Internally the calculator uses IEEE 754 double-precision floating point, which provides roughly 15 - 16 significant digits. Results are displayed to 4 decimal places. The practical precision of your result is limited by your input precision. If you measure angles to the nearest degree, trust results to about 2 - 3 significant figures. For surveying-grade accuracy, input angles in decimal degrees with at least 2 decimal places.

The circumradius R is the radius of the circle passing through all three vertices (circumscribed circle). The inradius r is the radius of the largest circle fitting inside the triangle (inscribed circle), tangent to all three sides. The circumradius is used in navigation, satellite positioning, and optics. The inradius appears in structural engineering for calculating the maximum inscribed bolt pattern or pipe diameter. The relation r = (Area) ÷ s provides a quick sanity check on computed values.