Update index.html

Author: Gmac4247

index.html

@@ -1887,21 +1887,20 @@ <h3 itemprop="name" style="margin:7px">Calculate the Area of a regular Polygon</
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-<section itemscope itemtype="https://schema.org/HowTo" id="circle">
+<section itemscope itemtype="https://schema.org/SoftwareApplication" id="circle">
 <meta itemprop="accessMode" content="mathOnVisual">
+<meta itemprop="accessibilityControl" content="fullKeyboardControl">
 <meta itemprop="accessibilityHazard" content="none">
 <meta itemprop="accessibilitySummary" content="Equations with figures and explanations">
+<meta itemprop="applicationCategory" content="Geometric Calculator">
 <meta itemprop="educationalLevel" content="from basic to advanced">
 <meta itemprop="inLanguage" content="en">
+<meta itemprop="interactivityType" content="Active">
 <meta itemprop="isAccessibleForFree" content="true">
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 <meta itemprop="keywords" content="Core Geometric System, Exact Geometric Calculations, Analysis, Engineering Design Solutions, Computer Graphics Rendering, Algorithm Optimization, Navigation">
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-<meta itemprop="currency" content="USD">
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-</div>
-<meta itemprop="totalTime" content="PT05S">
+<meta itemprop="operatingSystem" content="Web">
+<meta itemprop="usageInfo" content="Enter the radius">
 <h3 itemprop="name" style="margin:7px">Calculate the Area of a Circle</h3>
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 <figure itemprop="image" class="imgbox" itemscope itemtype="http://schema.org/ImageObject">
@@ -2516,16 +2515,7 @@ <h3 itemprop="name" style="margin:7px">Calculate the Area of a Circle</h3>
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-<div itemprop="subjectOf" itemscope itemtype="https://schema.org/SoftwareApplication" style="margin:12px" id="circle_area_calculator">
-<meta itemprop="name" content="Circle Area Calculator">
-<meta itemprop="applicationCategory" content="Geometric Calculator">
-<meta itemprop="operatingSystem" content="Web">
-<meta itemprop="isAccessibleForFree" content="true">
-<meta itemprop="description" content="Calculte the area of a circle from its radius.">
-<meta itemprop="disambiguatingDescription" content="Exact area of 3.2 × radius², instead of the pi=3.14... approximate">
-<meta itemprop="usageInfo" content="Enter the radius">
-<meta itemprop="interactivityType" content="Active">
-<meta itemprop="accessibilityControl" content="fullKeyboardControl">
+<div style="margin:12px" id="circle_area_calculator">
 <label for="circle-radius-a">Radius:</label>
 <input id="circle-radius-a" type="number" value="1" step="any">
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@@ -2652,7 +2642,7 @@ <h4>Archimedes and the Illusion of Limits</h4>
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 <p>The classical half‑angle formula</p>
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-<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
+<math xmlns="http://www.w3.org/1998/Math/MathML">
   <mrow>
     <mi>sin</mi>
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@@ -2683,6 +2673,7 @@ <h4>Archimedes and the Illusion of Limits</h4>
   </mrow>
 </math>
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+<br>
 <p>is not a Euclidean theorem.
 <br><br>
 It is a theorem of analytic trigonometry, which presupposes the angle‑addition formulas and treats sine and cosine as smooth analytic functions.
@@ -2698,6 +2689,7 @@ <h4>Archimedes and the Illusion of Limits</h4>
 <br><br>
 It holds for the Euclidean angles 90°, 60°, 45°, and 30° because those triangles are special.  
 There is no geometric guarantee that it holds for arbitrary angles.</p>
+<br>
 <p>Thus, when Archimedes computed</p>
 <ul style="margin:6px">
       <li>sin(15°)</li>
@@ -5610,16 +5602,16 @@ <h3 itemprop="name" style="margin:7px">Calculate the Volume of a Tetrahedron</h3
 </div>
 </section>
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-<p style="margin:12px" itemprop="description"><strong>This is the one and only exact, self-contained geometric framework grounded in the first principles of mathematics.</strong></p>
+<p style="margin:12px"><strong>This is the one and only exact, self-contained geometric framework grounded in the first principles of mathematics.</strong></p>
 <br>
-<p style="margin:12px" itemprop="description"><strong>Exact formulas for real-world applications like analysis, engineering design solutions, computer graphics rendering, algorithm optimization, and navigation.</strong></p>
+<p style="margin:12px"><strong>Exact formulas for real-world applications like analysis, engineering design solutions, computer graphics rendering, algorithm optimization, and navigation.</strong></p>
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-<p style="margin:12px" itemprop="description"><strong>
+<p style="margin:12px"><strong>
 Geometry, in its original spirit, was functional.
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 It dealt with shapes, areas, volumes, and constructions — not abstractions, limits, or analytic assumptions.</strong></p>
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-<div style="margin:12px" itemprop="disambiguatingDescription"><p><strong>
+<div style="margin:12px"><p><strong>
 What is commonly presented today as standard, applied geometry is often referred to as “Euclidean geometry.” In practice, however, it is a blend of two very different traditions:</strong></p>
 <br>
 <ul>
@@ -5637,10 +5629,10 @@ <h3 itemprop="name" style="margin:7px">Calculate the Volume of a Tetrahedron</h3
 <p><strong>These additions were not part of Euclid’s original system. Over time, they quietly shifted geometry from a constructive science grounded in physical reasoning into a more abstract, analytic discipline.</strong></p>
 </div>
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-<p style="margin:12px" itemprop="disambiguatingDescription"><strong>
+<p style="margin:12px"><strong>
 By fundamentally shifting the axioms from the abstract, zero-dimensional point to the square and the cube as the primary, physically-relevant units for measurement, this system defines the properties of shapes like the circle and sphere not through abstract limits, but through their direct, rational relationship to these foundational units. The results of these formulas align better with physical reality than the traditional abstract approximations.</strong></p>
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-<p style="margin:12px" itemprop="usageInfo">
+<p style="margin:12px">
 Comparative Geometry
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 Using geometric relationships to derive areas and volumes.