Triangle Calculator

Triangles: A complete guide to sides, angles, and area

Triangles are fundamental geometric figures defined by three sides and three angles. With the right inputs, you can solve all unknowns and derive area, perimeter, altitudes, and inscribed/circumscribed circle radii. This guide explains valid input cases and the formulas the calculator uses.

Valid input cases

• SSS: Three sides known. Use law of cosines to find angles, then compute area via Heron’s formula.
• SAS: Two sides and included angle known. Use law of cosines to find the third side, then law of sines for remaining angles.
• ASA/AAS: Two angles and one side known. Use angle sum to find the third angle, then law of sines for remaining sides.

Core formulas

• Law of cosines: \(c^2=a^2+b^2-2ab\cos C\), and cyclic permutations.
• Law of sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
• Heron’s area: \(A=\sqrt{s(s-a)(s-b)(s-c)}\), where \(s=\frac{a+b+c}{2}\).
• Trigonometric area: \(A=\frac{1}{2}ab\sin C\) for SAS.
• Altitudes: \(h_a=\frac{2A}{a}\), etc.
• Inradius: \(r=\frac{A}{s}\), Circumradius: \(R=\frac{a}{2\sin A}\) (or \(R=\frac{abc}{4A}\)).

Classification

• By sides: Equilateral (a=b=c), Isosceles (two equal), Scalene (all different).
• By angles: Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°).

Best practices

• Validate: Angles must sum to 180°; sides must satisfy the triangle inequality.
• Use degrees consistently: This calculator accepts degrees for all angles.
• Check rounding: Small rounding differences are normal; keep adequate precision in inputs.

Worked example