Universal Triangle Solver (SSS, SAS, ASA, AAS)

Enter any three parameters of the triangle (three sides, two sides and an angle, or two angles and a side) to find all missing values, the area, the perimeter, and the type of triangle.

Triangle Solving: Law of Sines vs. Law of Cosines

To solve a triangle (find all missing sides and angles), you need at least three pieces of information, including at least one side. The core of triangle solving relies on two trigonometric laws.

Law of Cosines (for SSS and SAS Cases)

The Law of Cosines is typically used first when you know all three sides (SSS) or two sides and the angle *between* them (SAS).

Law of Sines (for ASA, AAS, and to finish solutions)

The Law of Sines is used when you know a side and its opposite angle, or two angles and a side (ASA or AAS).

The **Sum of Angles** is always $180^\circ$: $\alpha + \beta + \gamma = 180^\circ$.

Area Calculation

The calculator uses the most efficient area formula based on your inputs:

• **SAS Case:** Area = $\frac{1}{2}ab\sin(\gamma)$
• **SSS Case:** Area is calculated using Heron's Formula (requires all three sides).

Frequently Asked Questions (FAQ)

What are the three ways to classify a triangle?

Triangles are classified in two ways:

• **By Sides:** Equilateral (3 equal sides), Isosceles (2 equal sides), Scalene (no equal sides).
• **By Angles:** Acute (all angles < 90°), Obtuse (one angle > 90°), Right (one angle = 90°).

Can a triangle have two right angles?

No. Since the sum of angles in any triangle must be $180^\circ$, having two right angles ($90^\circ + 90^\circ = 180^\circ$) would leave $0^\circ$ for the third angle, making it impossible to form a closed shape.

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a^2 = b^2 + c^2 - 2bc \cos(\alpha)

\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}

$ a^2 = b^2 + c^2 - 2bc \cos(\alpha) $

$ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} $