About

An acute triangle is defined by a strict constraint: every interior angle α, β, γ must satisfy θ < 90°. Equivalently, for sides a, b, c, the relation a² + b² > c² must hold for every permutation. Misidentifying a triangle as acute when it is obtuse leads to incorrect structural assumptions in truss engineering, roof pitch calculations, and computational geometry collision detection. This calculator validates the acute condition before computing results.

The tool computes area via Heron's formula, all three altitudes h_a, h_b, h_c, medians m_a, m_b, m_c, the inradius r, and circumradius R. It accepts six input modes (SSS, SAS, ASA, AAS, AAA+side, SSA) and resolves unknowns via the Law of Cosines and Law of Sines. Note: AAA mode requires at least one side length to determine scale. SSA may yield an ambiguous case; only the acute-valid solution is returned.

Formulas

The primary solving method depends on the input mode. For three known sides (SSS), all angles are recovered via the Law of Cosines:

α = arccos b² + c² − a²2bc

For cases involving angles, the Law of Sines provides cross-resolution:

asin α = bsin β = csin γ = 2R

Area is computed via Heron's formula once all three sides are known:

A = √s(s − a)(s − b)(s − c)

The acute triangle condition is verified by checking all three squared-side inequalities:

a² + b² > c² and a² + c² > b² and b² + c² > a²

Where: a, b, c = side lengths opposite to angles α, β, γ respectively. s = semi-perimeter. A = area. R = circumradius. r = inradius. h_a = altitude to side a. m_a = median to side a.

Reference Data

Property	Formula	Unit	Notes
Perimeter	P = a + b + c	length	Sum of all sides
Semi-perimeter	s = P2	length	Used in Heron's formula
Area (Heron)	A = √s(s−a)(s−b)(s−c)	length²	Numerically stable for all acute triangles
Area (Trig)	A = 12ab sin γ	length²	Any two sides and included angle
Altitude h_a	h_a = 2Aa	length	Perpendicular from vertex A to side a
Altitude h_b	h_b = 2Ab	length	Perpendicular from vertex B to side b
Altitude h_c	h_c = 2Ac	length	Perpendicular from vertex C to side c
Median m_a	m_a = 12√2b² + 2c² − a²	length	From vertex A to midpoint of a
Median m_b	m_b = 12√2a² + 2c² − b²	length	From vertex B to midpoint of b
Median m_c	m_c = 12√2a² + 2b² − c²	length	From vertex C to midpoint of c
Inradius	r = As	length	Radius of inscribed circle
Circumradius	R = a2 sin α	length	Radius of circumscribed circle
Angle α	α = arccos b² + c² − a²2bc	°	Opposite to side a; must be < 90°
Angle β	β = arccos a² + c² − b²2ac	°	Opposite to side b; must be < 90°
Angle γ	γ = 180° − α − β	°	Must be < 90°
Acute condition (sides)	a² + b² > c²	-	Must hold for all three permutations
Triangle inequality	a + b > c	-	Must hold for all three permutations
Angle sum	α + β + γ = 180°	°	Euclidean plane constraint

Frequently Asked Questions

After resolving all three sides, the calculator checks whether a² + b² > c² holds for every permutation of the sides. If any permutation yields equality, the triangle is right-angled. If any yields a² + b² < c², the triangle is obtuse. Only when all three inequalities are strict does it classify as acute. Equivalently, all computed angles α, β, γ must be strictly less than 90°.

The SSA configuration can produce zero, one, or two valid triangles (the ambiguous case of the Law of Sines). The calculator computes sin of the unknown angle. If sin > 1, no triangle exists. If exactly one solution yields all angles < 90°, that solution is returned. If both solutions produce acute triangles, the calculator returns the one where the unknown angle is smaller, since that is the principal arcsin value. A warning is displayed noting the ambiguity.

Three angles alone define a similarity class, not a unique triangle. Infinitely many triangles share the same angles but differ in scale. The calculator requires one side length to fix the scale factor. Without it, area, perimeter, altitudes, medians, inradius, and circumradius are all indeterminate. The tool validates that the three angles sum to exactly 180° before proceeding.

Once all three sides a, b, c and the area A are known: altitude hₐ = 2A/a; median mₐ = ½√(2b² + 2c² − a²); inradius r = A/s where s is the semi-perimeter; circumradius R = a/(2·sin α). These formulas are exact for Euclidean triangles and do not involve iterative approximation.

Yes. An equilateral triangle has all angles equal to 60°, which is less than 90°. It is the most symmetric acute triangle. Enter any three equal side lengths in SSS mode to verify: the calculator will confirm the acute classification and return three identical altitudes, medians, and 60° angles.

All internal computation uses IEEE 754 double-precision floating point (approximately 15-17 significant decimal digits). Results are displayed to 4 decimal places by default. Rounding errors become relevant for near-degenerate triangles where one angle approaches 0° or the triangle inequality is barely satisfied. The calculator warns when any angle is within 0.01° of 0° or 90°.