Geometrie & Funktionen · Geometry & Area Equivalence
Flächenzerlegung & Scherungs-Visualisierer
Geometrische Zerlegungsgleichheit & Scherung: Parallelogramm, Dreieck und Flächeninvarianz
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Experimentelle Steuerungsparameter
Typ
Sektoren
cm
cm
cm
Echtzeit-Herleitung & Kennzahlen
Theoretische Fläche
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Exact analytical area
Grundseite des Ersatzrechtecks
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Circle unrolls to half-circumference πr
Höhe des Ersatzrechtecks
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Corresponds to radius r or height h
Umfangszunahme
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Perimeter increases by 2r due to two exposed radial sides
Echtzeit-Formelherleitung
Grundformel:
--Einsetzungsschritte:
--Ergebnis: --
Typische Beispielaufgaben & Prüfungsfallen
Circle-to-Rectangle Perimeter Increase & Area Calculation
Aufgabe: 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.
Analysestrategie: 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.
Schrittweise Herleitung:
- 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.
⚠️ Häufige Fehlvorstellung: Mistaking perimeter increase for area change; area is strictly conserved during rearrangement, while perimeter increases by exactly 2r.
Arbeitsblatt für den Unterricht & Hausaufgaben
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)
Häufige Fragen & typische Missverständnisse
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.