Triangle Angle Calculator

What Is a Triangle Angle Calculator?

A triangle angle calculator determines all unknown angles and sides of a triangle from a sufficient set of known values. Given any combination — three sides (SSS), two sides and an included angle (SAS), two angles and a side (ASA or AAS), two sides and a non-included angle (SSA), or a right triangle — the tool computes every measurement: all three angles, all three sides, area, perimeter, heights, medians, angle bisectors, inradius, and circumradius.

The Six Input Modes Explained

SSS — Three Sides Known

When all three side lengths a, b, and c are available, the Law of Cosines computes each angle. The tool verifies the triangle inequality (a + b > c, and so on) before computing. SSS uniquely determines the triangle — there is always exactly one solution.

A = arccos((b² + c² − a²) / (2bc))

SAS — Two Sides and the Included Angle

Given sides b and c with angle A between them, the missing side a is found via the Law of Cosines: a² = b² + c² − 2bc · cos(A). Angles B and C follow from A + B + C = 180°. SAS yields exactly one solution.

ASA and AAS — Two Angles and a Side

Angle C is derived from C = 180° − A − B, then the Law of Sines scales the remaining sides. ASA uses the side between the two angles; AAS uses the side opposite one of them. Both modes produce a single solution.

a / sin(A) = b / sin(B) = c / sin(C) = 2R

SSA — The Ambiguous Case

The most nuanced mode. Given angle A, opposite side a, and adjacent side b, the value sin(B) = b · sin(A) / a can yield up to four outcomes:

No triangle — if sin(B) > 1 (side a is too short to close the triangle).
One right triangle — if sin(B) = 1 exactly, so B = 90°.
One triangle — if sin(B) < 1 and a ≥ b.
Two triangles — if sin(B) < 1 and a < b; both the acute and obtuse interpretations of B produce a valid remaining angle C.

When two SSA solutions exist, the calculator displays both in separate tabs labelled Solution 1 (acute B) and Solution 2 (obtuse B) so you can compare them and choose the geometrically valid one.

Right-Triangle Mode (C = 90°)

Fixes angle C at 90° and accepts any two of the five remaining unknowns: leg a, leg b, hypotenuse c, angle A, or angle B. The Pythagorean theorem and basic trigonometric identities resolve the full triangle.

a² + b² = c²
sin(A) = a/c · cos(A) = b/c · tan(A) = a/b
R = c/2 · r = (a + b − c)/2

Derived Triangle Properties

Once all three sides are known, the calculator derives a full set of geometric properties automatically.

Area — Heron's Formula

s = (a+b+c)/2 Δ = √(s · (s−a) · (s−b) · (s−c))

Altitudes

hₐ = 2Δ/a h_b = 2Δ/b h_c = 2Δ/c

Medians and Angle Bisectors

mₐ = 0.5 · √(2b² + 2c² − a²)
tₐ = 2√(bc · s · (s−a)) / (b+c)

Inradius and Circumradius

r = Δ/s R = abc/(4Δ)

Law of Sines vs Law of Cosines

The Law of Cosines (a² = b² + c² − 2bc cos A) applies when at least two sides are known with an included angle (SAS) or when all three sides are known (SSS). It generalises the Pythagorean theorem and is numerically stable across all triangle shapes.

The Law of Sines suits cases with at least one angle–side opposite pair (ASA, AAS, SSA). When solving SSA, both the acute and obtuse interpretations of arcsin must be tested, since arcsin returns only the principal value in [0°, 90°].

Triangle Classification

By angles: acute (all < 90°), right (largest = 90°), or obtuse (largest > 90°) — determined by comparing a² + b² vs c² where c is the longest side.
By sides: equilateral (a = b = c), isosceles (any two sides equal within tolerance), or scalene (all sides different).

Degrees vs Radians

All internal calculations use radians for maximum precision. Input and output can be switched between degrees and radians at any time. One full turn equals 360° = 2π rad:

radians = degrees × π / 180

Radians are preferred in calculus, physics, and programming; degrees are the conventional choice in navigation and everyday geometry.

Practical Applications

Triangle angle calculators are used across many fields. Surveyors employ the Law of Sines to compute distances across impassable terrain using only angle readings. Structural engineers verify truss angles to ensure load paths remain within safe limits. Navigation systems triangulate positions using bearing and distance data. Game developers use trigonometric identities to compute lighting normals and collision vectors. Even everyday tasks — cutting mitre joints, laying floor tiles at an angle, or positioning a ladder safely — rely on right-triangle calculations.

Tips for accurate results: Use consistent units for all side lengths. In SSA mode, always verify both solutions and discard the physically impossible one. For very flat or very obtuse triangles, increase decimal precision to 6–8 places to avoid rounding artefacts in derived quantities.