# Law of Cosines Visualization

Interactive visualization of the Law of Cosines with draggable triangle and step-by-step proof

> Canonical page: https://elysiatools.com/en/visualizations/law-of-cosines

- **Category:** Math

## Overview

Interactive visualization of the Law of Cosines with draggable triangle vertices. Features four comprehensive sections: (1) Explore - Drag vertices A, B, C to change triangle shape in real-time. Displays side lengths (a, b, c) and angles (A, B, C) with live measurements. Shows the Law of Cosines formula c² = a² + b² - 2ab·cos(C) with substituted values. Automatically identifies triangle type (acute/right/obtuse) and provides step-by-step verification showing both sides of the equation are equal. (2) Proof - Animated 5-step derivation from Pythagorean Theorem. Step 1: Drop altitude h from B to AC, creating two right triangles. Step 2: Apply Pythagorean theorem to both triangles (h² = a² - x² and h² = c² - (b-x)²). Step 3: Equate expressions and expand (a² - x² = c² - b² + 2bx - x²). Step 4: Solve for x ((a² - c² + b²) / 2b). Step 5: Use cosine definition (x = a·cos(A)) to derive final formula c² = a² + b² - 2ab·cos(C). Each step includes visual diagram updates and mathematical explanations. (3) Examples - Pre-configured triangle demonstrations: Acute (all angles < 90°), Right (one angle = 90°, reduces to Pythagorean theorem), Obtuse (one angle > 90°, cos(C) is negative), SAS case (given two sides and included angle), SSS case (given three sides, find angles). Each example includes detailed explanations of how the Law of Cosines applies. (4) Practice - Two modes: Find Missing Side (given SAS) or Find Missing Angle (given SSS). Random problem generator with valid triangle constraints. Interactive canvas visualization of each problem. Input validation with answer checking. Step-by-step solution reveal feature. Practice history tracking showing recent attempts with correctness indicators. Real-time formula substitution and calculation display. Educational content covers relationship to Pythagorean theorem, behavior in different triangle types, and practical applications. Multi-language support (zh, en, es, fr, de, ru, pt) with comprehensive translations. Responsive design with color-coded measurements, smooth animations, and mobile-friendly touch controls.

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