Geometría interactiva y funciones · Geometry & Area Equivalence
Visualizador de descomposición y conservación de áreas
Visualizador geométrico de corte y pegado de figuras: principio de conservación del área
- Cálculo en cliente
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- Derivación en vivo
- Modo proyector
- Hoja de trabajo A4
Ventana de demostración geométrica● 60 FPS
Variables de control del experimento
tipo
sectores
cm
cm
cm
Derivaciones y métricas en tiempo real
Área teórica
--
Exact analytical area
Base del rectángulo equivalente
--
Circle unrolls to half-circumference πr
Altura del rectángulo equivalente
--
Corresponds to radius r or height h
Aumento del perímetro
--
Perimeter increases by 2r due to two exposed radial sides
Derivación de fórmulas en tiempo real
Fórmula básica:
--Sustitución de valores:
--Conclusión: --
Ejemplos clásicos y análisis de errores en exámenes
Circle-to-Rectangle Perimeter Increase & Area Calculation
Problema: A circular paper is cut into equal sectors and rearranged into an approximate rectangle. The perimeter of the rectangle is 10 cm greater than the circle's original circumference. Find the area of the circle.
Estrategia analítica: The two long sides sum to 2 × (πr) = 2πr (the circle's perimeter). The additional perimeter comes entirely from the two short ends, which each equal the radius r. Thus ΔC = 2r = 10 cm, giving r = 5 cm.
Derivación paso a paso:
- Step 1: Let the circle radius be r. Circle circumference C = 2πr.
- Step 2: Equivalent rectangle length = πr, width = r. Rectangle perimeter = 2(πr + r) = 2πr + 2r.
- Step 3: Perimeter difference ΔC = 2r = 10 cm, so r = 5 cm.
- Step 4: Compute area S = πr² = π × 5² = 25π ≈ 78.54 cm².
⚠️ Error conceptual frecuente: Mistaking perimeter increase for area change; area is strictly conserved during rearrangement, while perimeter increases by exactly 2r.
Hoja de indagación en clase y tareas
Q1. A circle with radius 6 cm is cut into 32 sectors to form a rectangle. By how many cm does the perimeter increase?
Answer Key: 12 cm (exactly 2r = 2 × 6)
Q2. A parallelogram (base 10 cm, height 6 cm) is cut along height and reshaped into a rectangle. How do area and perimeter change?
Answer Key: Area unchanged (60 cm²), perimeter decreases (slanted edge replaced by shorter vertical height)
Preguntas frecuentes y errores conceptuales
Why are the top and bottom edges of the reassembled circle slightly wavy?
Each sector's outer edge is a circular arc. As slice count increases from 16 to 64 and infinity, the arc curvature flattens, and the boundary converges strictly to parallel line segments.