Universal Triangle Engine

Triangle AI Solver: Step-by-Step Solutions

Solve any triangle (SSS, SAS, ASA) or prove triangle theorems. Type your problem, upload a photo, or use a template below.

b (base) c a α β γ

Quick Start Templates

How to Solve Any Triangle Problem with AI

From basic geometry to advanced trigonometry

90°

Right-Angled Triangles

Our engine instantly recognizes right triangles. It applies the Pythagorean Theorem ($a^2 + b^2 = c^2$) for sides and SOH CAH TOA for angles.

  • • Hypotenuse calculation
  • • Leg finding
  • • Inverse trigonometry
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Oblique Triangles

For non-right triangles, the solver intelligently selects between the Law of Sines and the Law of Cosines depending on the given SSA, SAS, or SSS data.

  • • Ambiguous Case (SSA) detection
  • • Heron’s Formula for Area
  • • Missing angle solving
QED

Triangle Proofs

Need to prove congruence or similarity? The AI generates formal Two-Column Proofs. It identifies properties like Reflexive Property, CPCTC, and vertical angles.

  • • SSS, SAS, ASA, AAS, HL
  • • Isosceles Triangle Theorems
  • • Midsegment Theorem

Comprehensive Triangle Formulas Guide

Solving triangles requires selecting the right formula based on the “Given” values. Our AI automatically determines which method to use, but understanding the underlying logic helps you learn. Here are the core formulas used by our engine.

Trigonometric Laws

Law of Sines (AAS, ASA, SSA)

$$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$

Law of Cosines (SSS, SAS)

$$ c^2 = a^2 + b^2 – 2ab \cos C $$

Area Calculation Methods

Standard Formula (Base & Height)

$$ Area = \frac{1}{2} \cdot \text{base} \cdot \text{height} $$

Heron’s Formula (SSS)

$$ Area = \sqrt{s(s-a)(s-b)(s-c)} $$

where $s = \frac{a+b+c}{2}$

Triangle Congruence & Similarity Guide

Proving that two triangles are congruent (identical in shape and size) or similar (identical in shape, different in size) is a core part of geometry. Our AI solver recognizes these standard postulates:

  • SSS Side-Side-Side: All three sides of one triangle are equal to the three sides of another.
  • SAS Side-Angle-Side: Two sides and the included angle are equal.
  • ASA Angle-Side-Angle: Two angles and the included side are equal.
  • AAS Angle-Angle-Side: Two angles and a non-included side are equal.
  • HL Hypotenuse-Leg: Specifically for right triangles, if the hypotenuse and one leg are equal.

Frequently Asked Questions

Can it solve for altitude and medians?

Yes. Once the triangle is solved (all 3 sides known), the engine can calculate secondary segments like altitude (height), medians, and angle bisectors.

Does it support Heron’s formula?

Absolutely. If you provide three side lengths (SSS), the solver automatically uses Heron’s Formula ($A = \sqrt{s(s-a)(s-b)(s-c)}$) to find the area without needing the height.

What if I have the “Ambiguous Case” (SSA)?

The AI checks for the Ambiguous Case. If two solutions exist (two possible triangles), it will calculate and display both sets of values.