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Update HTML structure for semantic clarity
Refactor HTML structure by replacing <div> elements with <section> and <article> tags for better semantic organization.
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about.html

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<button onclick="document.getElementById('cookie-notice').style.display = 'none';">Reject</button>
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<h1 style="font-size:160%;margin:7px;">How Accurate Are The Conventional Geometry Formulas?</h1>
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<p style="margin:12px;">Historically, Euclidean geometry has provided a framework for understanding and describing the physical world. It is based on axioms and postulates, leading to well-defined formulas for the calculation of areas and volumes of shapes such as circles and spheres.
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<p style="margin:7px;"><strong>Rethinking the Circle: A Logical Reexamination of Area and Circumference</strong>
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<p style="margin:12px;"><strong>Rethinking the Circle: A Logical Reexamination of Area and Circumference</strong>
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For centuries, the circle has been a symbol of mathematical elegance—and π its most iconic constant. But beneath the surface of tradition lies a deeper question: Are the formulas we use truly derived from geometric logic, or are they inherited approximations dressed in symbolic authority?
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<p style="margin:12px;"><b>Historical Approximations: Respect, Not Reverence</b>
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Historical records suggest that ancient Babylonians initially calculated it as 3, later they used 3.125; Egyptians estimated it is ( 16 / 9 )² ~ 3.16.
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<p style="margin:12px;"><b>Archimedes and the Polygonal Trap</b>
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<p style="margin:12px;"><b>The Symbol π: A Linguistic Shortcut</b>
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It was not until the 18th century that the symbol π, popularized by the mathematicians of the time, gained widespread acceptance.
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<p style="margin:12px;"><b>∫ Calculus: Summary, Not Source</b>
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It can be exact with exact limits and basic operations, but if those are given then they can be calculated directly without calculus.
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<p style="margin:12px;"><b>The golden ratio</b>
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<p style="margin:12px;"><b>φ The golden ratio:</b>
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Some relate the numeric value of 3.14… to the so-called “golden ratio” of ( √5 + 1 ) / 2.
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That has no logical ties to the area nor the circumference of a circle.
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<b>A Rational Alternative: 3.2</b>
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These values are exact, rational, and logically derived. They can be verified numerically, but more importantly, they can be proven algebraically—without relying on infinite fractions, symbolic shortcuts, or flawed assumptions.
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<strong style="font-size:160%;margin:7px;">THE AREA OF A CIRCLE is defined by comparing it to a square since that is the base of area calculation.</strong>
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When the arcs of the quadrant circles intersect at the quarter of the centerline of the square, the uncovered area in the middle equals exactly the sum of the overlapping areas respectively.
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<p style="margin:12px;">The quadrant method not only proves that the area of a circle is 3.2 × radius², it necessarily rules out the validity of the π.
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The area of a circle is exactly 3.2 × radius².
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<strong style="font-size:160%;margin:7px;">THE CIRCUMFERENCE OF A CIRCLE can be derived from the area algebraically.</strong>
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<mi>r</mi>
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Irrational or not, with an infinitesimally small thickness the circumference practically equals 6.4 × radius.
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Since the ratio can be expressed as a real number, there is no reason to substitute it with any other sign.
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The best practice is writing it as it is.
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From a historical perspective the mathematical constant named π is what it is. It's unlikely to change much.
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<p style="margin:12px;"><b>Conclusion: Time to Move On</b>
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From a scientific perspective they call it irrational for a reason. It doesn't make much sense. It's a logical dead end.
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The π has served its symbolic purpose. But in geometry, clarity matters more than tradition.
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The people deserve better than an irrational approximation if an exact calculation is available.
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Since the true ratio is exactly 3.2, and that is a rational number, then we can—and should—write it as it is. Let the π remain in the history books. Geometry deserves better.
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That makes the arc value of 360° = 6.4radian, and
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sin( π / 2 ) < 1 .
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That makes the arc value of 360° = 6.4radian, and trigonometric functions that rely on arc value have to be aligned to 3.2 respectively.
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These are two aspects of that.
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<strong style="font-size:160%;margin:7px;">THE VOLUME OF A SPHERE is another aspect of the area relationship, cubing the square root of its cross sectional area.
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If you're trying to calculate the volume of a physical ball or sphere for a practical purpose – whether it's for a science experiment, engineering, or any other real-world application – my empirically derived V = " cubic value of ( √( 3.2 ) × radius ) " formula offers a result that aligns more closely with what you would measure in the lab.
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<strong style="font-size:160%;margin:7px;">THE VOLUME OF A CONE can be derived algebraically by comparing a vertical quadrant of a cone to an octant of a sphere.
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<p style="margin:12px;">The commonly used base × height / 3 approximation for the volume of a pyramid was likely estimated based on two observations.
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No. Because it's not true in case of most other shapes.
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<p style="margin:12px;">The other idea is the cube dissection.
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Also it's not just about the vertices, but the edges and the inner faces, too.
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</article>
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