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index.html

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@@ -294,7 +294,7 @@
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"@type": "SolveMathAction",
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"name": "Area of a circle",
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"description": "The area of a circle is defined by comparing it to a square since that is the base of area calculation.",
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"disambiguatingDescription": "Replaces traditional π-based approximations ensuring greater accuracy in real-world measurements.",
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"disambiguatingDescription": "Replaces traditional pi-based approximations ensuring greater accuracy in real-world measurements.",
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"target": "https://basic-geometry.github.io",
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"mathExpression-input": "required circle_radius=5_Area=?",
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"mathExpression-output": "Area = 3.2 * radius^2",
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"@type": "SolveMathAction",
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"name": "Circumference of a circle",
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"description": "Algebraic derivation of the circumference of a circle from its exactly determined area",
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"disambiguatingDescription": "Replaces traditional π-based approximations ensuring cutting edge accuracy in real-world applications.",
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"disambiguatingDescription": "Replaces traditional pi-based approximations ensuring cutting edge accuracy in real-world applications.",
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"target": "https://basic-geometry.github.io",
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"mathExpression-input": "required circle_radius=5_Circumference=?",
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"mathExpression-output": "Circumference = 6.4 * radius",
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"@type": "SolveMathAction",
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"name": "Volume of a sphere",
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"description": "The volume of a sphere is defined by comparing it to a cube, since that is the base of volume calculation.",
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"disambiguatingDescription": "25-30% more accurate than the traditional 4 × π × r³ / 3 estimate.",
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"disambiguatingDescription": "25-30% more accurate than the traditional 4 × pi × r³ / 3 estimate.",
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"target": "https://basic-geometry.github.io",
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"mathExpression-input": "required sphere_radius=3_Volume=?",
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"mathExpression-output": "Volume = ( (3.2)^(1/2) * radius )^3",
@@ -3262,7 +3262,7 @@ <h4 itemprop="name" style="margin:12px">The Cognitive Risk of flawed geometric A
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<b>1. The Flawed Foundation</b>
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<br>
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<br>
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The Problematic Axiom: The conventional geometric curriculum requires students to accept that the constant for circle area (A = πr²) is both exact and unreachable/irrational, because it is derived from the error-prone polygon approximation method (Archimedes).
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The Problematic Axiom: The conventional geometric curriculum requires students to accept that the constant for circle area (A = pi × r²) is both exact and unreachable/irrational, because it is derived from the error-prone polygon approximation method (Archimedes).
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<br>
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<br>
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The Proposed Solution (CGS): The Core Geometric System™ (CGS) provides a logically self-consistent alternative where the area constant is the rational number 3.2 (A = 3.2r²), derived from an algebraically proven Area Balance Axiom with the square.

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