Triangle Solver

By angles

• Acute — all three angles under 90°
• Right — exactly one angle is 90°
• Obtuse — one angle over 90°

By sides

• Equilateral — all three sides equal (and all angles equal 60°)
• Isosceles — exactly two sides equal (the angles opposite those sides are equal too)
• Scalene — no sides equal, no angles equal

Every triangle gets one label from each list, e.g. “Right · Scalene” or “Obtuse · Isosceles.”

Vocabulary

• Vertex (plural vertices) — a corner point
• Side — the general term for any of a triangle’s three edges, in every type of triangle
• Hypotenuse — a right triangle’s longest side, opposite the right angle. This name only applies to right triangles
• Legs — a right triangle’s other two sides (the ones forming the right angle). Also only a right-triangle term — in any other triangle, all three edges are just called “sides”
• Altitude — the perpendicular distance from a vertex to the line of the opposite side (see the Alt. rows above). For an obtuse triangle, two of the three altitudes land outside the triangle itself, on an extension of that side

Any three of the six values (three sides, three angles) fix a triangle’s exact size and shape — except three angles alone (AAA), which only fix its shape, not its size.

Law of Cosines (used when all three sides are known, or two sides + the angle between them):

a² = b² + c² − 2bc·cos(A)

Law of Sines (used when two angles are known, or when checking the ambiguous two-sides-plus-non-included-angle case):

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

The ambiguous case (SSA): two sides plus an angle that is not between them can produce zero, one, or two valid triangles. If two exist, both are shown as tabs above.

Area: computed via Heron’s formula from the three sides once known.