About

The ASA (Angle-Side-Angle) configuration defines a triangle uniquely when two angles and the included side are known. The third angle B is computed as 180° − A − C. The remaining sides are resolved through the Law of Sines. Incorrect angle entry where A + C ≥ 180° produces a degenerate or impossible figure. This tool computes all six primary elements plus derived quantities: three altitudes, three medians, inradius r, circumradius R, and area S. Results assume Euclidean plane geometry. Accuracy degrades for near-degenerate triangles where any angle approaches 0° or 180°.

Formulas

The ASA method begins by computing the missing angle from the triangle angle sum:

B = 180° − A − C

The Law of Sines then resolves the unknown sides:

asin A = bsin B = csin C

Therefore:

a = b ⋅ sin Asin B

c = b ⋅ sin Csin B

The area uses the trigonometric formula:

S = 12 ⋅ a ⋅ c ⋅ sin B

The circumradius R and inradius r are:

R = a2 sin A

r = Ss

Where A = angle at vertex A, B = angle at vertex B, C = angle at vertex C, a = side opposite A, b = side opposite B (the known included side), c = side opposite C, S = area, s = semi-perimeter, R = circumradius, r = inradius.

Reference Data

Property	Formula	Description
Third Angle	B = 180° − A − C	Angle sum property of triangles
Side a	a = b ⋅ sin Asin B	Law of Sines - side opposite angle A
Side c	c = b ⋅ sin Csin B	Law of Sines - side opposite angle C
Perimeter	P = a + b + c	Sum of all sides
Semi-perimeter	s = P2	Half the perimeter, used in many formulas
Area (Trig)	S = 12 ⋅ a ⋅ c ⋅ sin B	Area from two sides and included angle
Area (Heron)	S = √s(s−a)(s−b)(s−c)	Heron's formula - cross-check
Height h_a	h_a = 2Sa	Altitude to side a
Height h_b	h_b = 2Sb	Altitude to side b
Height h_c	h_c = 2Sc	Altitude to side c
Median m_a	m_a = 12√2b² + 2c² − a²	Median to side a
Median m_b	m_b = 12√2a² + 2c² − b²	Median to side b
Median m_c	m_c = 12√2a² + 2b² − c²	Median to side c
Inradius	r = Ss	Radius of inscribed circle
Circumradius	R = a2 sin A	Radius of circumscribed circle
Angle Bisector t_a	t_a = 2bcb + c ⋅ cos A2	Bisector from vertex A
Triangle Type (sides)	-	Equilateral, Isosceles, or Scalene
Triangle Type (angles)	-	Acute, Right, or Obtuse

Frequently Asked Questions

A valid Euclidean triangle requires all three interior angles to sum to exactly 180°. If A + C ≥ 180°, the third angle B would be zero or negative, which is geometrically impossible. The calculator rejects such inputs and displays an error.

Two angles fix the shape (all angles are determined since B = 180° − A − C), and the included side fixes the scale. Unlike the SSA (ambiguous) case, ASA cannot produce two distinct triangles. This is guaranteed by the Angle-Side-Angle congruence theorem.

When either angle approaches 0° or the third angle approaches 180°, the triangle becomes extremely elongated. Floating-point precision in sin calculations near 0 leads to disproportionately large side values. The tool warns when any angle is below 1° or above 178°.

The input fields accept degrees by default. Internally, values are converted to radians via θ_rad = θ_deg ⋅ π180 before any trigonometric function is called. A unit toggle allows switching to radian input if needed.

The canvas computes vertex coordinates by placing vertex A at the origin, vertex B along the positive x-axis at distance c, then vertex C at coordinates (b cos A, b sin A). The figure is then scaled and centered to fit the canvas dimensions while preserving the true proportions. All labels are positioned to avoid overlap.

In ASA, the known side is between the two known angles. In AAS, the known side is opposite one of the known angles. Both uniquely determine the triangle. Mathematically, knowing any two angles automatically gives the third, so ASA and AAS reduce to the same computation. This calculator handles both by always computing the third angle first.