Geometría interactiva y funciones · Common Core Grade 8 Geometry (Pythagorean Theorem Applications), GCSE Maths (3D Geodesics & Unfolding), AMC 8/10 Surface Geometry.

Visualizador de trayectoria mínima y desarrollo de poliedros

Q: Why does a 2-turn spiral on a cylinder correspond to twice the circumference in the unfolded rectangle?

Each full turn traverses one unrolled cylinder width. Two turns are equivalent to tiling two identical rectangles side by side, making the horizontal base exactly 2 × (2πr).

Visualizador de distancia mínima en superficies 3D mediante desarrollo plano de cilindros y cubos

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Derivación en vivo
Modo proyector
Hoja de trabajo A4

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Ventana de demostración geométrica● 60 FPS

Velocidad:

Preajustes y errores frecuentes:

Variables de control del experimento

Sólido (0 cilindro, 1 ortoedro, 2 cono)

tipo

Radio r / longitud a

Altura h / ancho b

Altura c (solo ortoedro)

Vueltas (solo cilindro)

vueltas

Derivaciones y métricas en tiempo real

Camino mínimo por la superficie L

Straight line distance on unfolded net

Base horizontal desarrollada

2πr × turns for cylinder

Camino 1 del ortoedro

Unfolding front-top face pair

Camino 2 del ortoedro

Unfolding front-right face pair

Derivación de fórmulas en tiempo real

Fórmula básica: --

Sustitución de valores: --

Conclusión: --

Versión para primaria:🎮 Desarrollo, área y volumen del ortoedro ↗

Ejemplos clásicos y análisis de errores en exámenes

Cuboid Geodesic: Dimensions 3×4×5, Shortest Path from Corner A to Opposite Corner B

Problema: A cuboid has length 3 cm, width 4 cm, height 5 cm. An ant crawls on the outer surface from bottom corner A to opposite top corner B. Find the shortest path distance.

Estrategia analítica: Three unfolding pairs: (1) Combine length and width: √((3+4)² + 5²) = √74 ≈ 8.60 cm; (2) Combine length and height: √((3+5)² + 4²) = √80 ≈ 8.94 cm; (3) Combine width and height: √((4+5)² + 3²) = √90 ≈ 9.49 cm. Shortest path is √74 cm. General rule: sum the two smallest dimensions!

Derivación paso a paso:

Step 1: Sort dimensions: a=3 < b=4 < c=5.
Step 2: Unfolding 1: √((3+4)² + 5²) = √74 ≈ 8.60 cm.
Step 3: Unfolding 2: √((3+5)² + 4²) = √80 ≈ 8.94 cm.
Step 4: Unfolding 3: √((4+5)² + 3²) = √90 ≈ 9.49 cm.
Step 5: Compare: minimum is √74 ≈ 8.60 cm.

⚠️ Error conceptual frecuente: Testing only one unfolding: always compare all 3 permutations or use the golden rule that summing the two smallest dimensions yields the minimal hypotenuse.

Hoja de indagación en clase y tareas

Q1. Cylinder has circumference 12 cm and height 5 cm. What is the shortest surface path for 1 full turn around the side?

Q2. For a 2×3×4 cuboid, which two side lengths should be added together for the shortest surface path?

Preguntas frecuentes y errores conceptuales

Why does a 2-turn spiral on a cylinder correspond to twice the circumference in the unfolded rectangle?

Each full turn traverses one unrolled cylinder width. Two turns are equivalent to tiling two identical rectangles side by side, making the horizontal base exactly 2 × (2πr).