About

Calculating triangle area from two sides and their included angle (SAS) relies on the formula A = 12 ⋅ a ⋅ b ⋅ sin(C). This is the only reliable method when base-height pairs are unknown but a protractor or angle measurement is available. Misidentifying the included angle (using an adjacent angle instead) produces a silently wrong result. The formula degenerates when C = 0° or C = 180°, where the triangle collapses to a line segment.

This calculator computes area and then derives the full triangle solution: third side via the Law of Cosines, remaining angles via the Law of Sines, perimeter, all three altitudes, inradius r, and circumradius R. Results assume Euclidean plane geometry. For spherical or hyperbolic surfaces, these formulas do not apply. Note: floating-point arithmetic limits precision to approximately 12 significant digits.

Formulas

The SAS area formula computes the area of a triangle when two sides and their included angle are known.

Area = 12 ⋅ a ⋅ b ⋅ sin(C)

Where a and b are the two known sides, and C is the angle between them (the included angle).

The third side is recovered via the Law of Cosines:

c = √a² + b² − 2ab ⋅ cos(C)

The remaining angles are found using the Law of Sines:

sin(A)a = sin(C)c

Derived quantities use the following relations. The semi-perimeter:

s = a + b + c2

The inradius (radius of the inscribed circle):

r = Areas

The circumradius (radius of the circumscribed circle):

R = c2 ⋅ sin(C)

Each altitude (height relative to each side):

h_a = 2 ⋅ Areaa

Where s = semi-perimeter, r = inradius, R = circumradius, h_a = altitude to side a.

Reference Data

Triangle Type	Angle Condition	Side Relationship	Area Behavior	Notes
Equilateral	A = B = C = 60°	a = b = c	A = √34 ⋅ a²	Maximum symmetry
Isosceles	Two angles equal	Two sides equal	Standard SAS formula	Axis of symmetry bisects the unequal side
Scalene	All angles different	All sides different	Standard SAS formula	Most general case
Right (90°)	One angle = 90°	c² = a² + b²	A = 12ab (since sin90° = 1)	SAS simplifies to half-product of legs
Obtuse	One angle > 90°	Longest side opposite obtuse angle	Standard SAS formula	sin still positive for 0° - 180°
Acute	All angles < 90°	No side dominates	Standard SAS formula	Circumcenter lies inside triangle
Degenerate	C = 0° or 180°	Collinear points	Area = 0	Not a valid triangle
Golden gnomon	36° - 36° - 108°	Side ratio involves φ	Standard SAS formula	Related to Penrose tiling
30-60-90	30°, 60°, 90°	1 : √3 : 2	Standard SAS formula	Common in construction and drafting
45-45-90	45°, 45°, 90°	1 : 1 : √2	A = 12a²	Isosceles right triangle
Key Constants
π radians	= 180°		Conversion factor: 1° = π÷180 rad
sin(30°)	= 0.5		Exact value
sin(45°)	= √22 ≈ 0.7071		Exact value
sin(60°)	= √32 ≈ 0.8660		Exact value
sin(90°)	= 1		Maximum value of sine

Frequently Asked Questions

The SAS formula Area = ½·a·b·sin(C) requires C to be the angle formed at the vertex where sides a and b meet. If you use a non-included angle, the sine value does not correspond to the perpendicular height derived from those two specific sides, producing an incorrect area. Two sides and a non-included angle can sometimes define two different triangles (the ambiguous SSA case), making the result non-unique.

When C = 90°, sin(90°) = 1, so the formula simplifies to Area = ½·a·b. This is the familiar right-triangle area formula where the two given sides are the legs. The Law of Cosines also reduces to the Pythagorean theorem: c = √(a² + b²).

Yes. This calculator supports both units via a toggle. Internally, JavaScript's Math.sin() expects radians, so degree inputs are converted by multiplying by π/180. If you enter radians, ensure your value is in the range (0, π), which corresponds to (0°, 180°) for a valid triangle.

JavaScript uses IEEE 754 double-precision floats with about 15-17 significant decimal digits. For typical engineering or academic inputs (sides up to millions of units, angles away from 0° and 180°), errors remain below 10⁻¹⁰ relative. Near-degenerate triangles (angle close to 0° or 180°) amplify rounding error because sin(C) approaches 0 and the third side computation involves catastrophic cancellation.

After computing the third side c via the Law of Cosines, angle A is found from sin(A) = a·sin(C)/c. In the SAS case there is no ambiguity: the triangle is fully determined. The arcsine could theoretically return the wrong quadrant, but since the included angle C is known and the triangle is unique, angle A is checked against the constraint A + B + C = 180° to ensure correctness. Angle B is simply 180° − A − C.

The area ½·a·b·sin(C) is maximized when sin(C) = 1, i.e., C = 90°. The maximum area equals ½·a·b. As the angle moves away from 90° toward 0° or 180°, the area decreases toward zero. This is why a right angle produces the fattest triangle for given leg lengths.