Update index.html

Author: Gmac4247

index.html

@@ -286,10 +286,7 @@
 	"abstract": "The area of a circle is defined by comparing it to a square since that is the base of area calculation. The circle can be cut into four quadrants, each placed with their origin on the vertices of a square. In this layout the arcs of the quadrants of an inscribed circle would meet at the midpoints of the sides of the square. The arcs of the quadrants of a circumscribed circle would meet at the center of the square. The arcs of the quadrants that equal in area to the square intersect right in between these limits on its centerlines. The ratio between the radius of the circle and the side of the square is calculable. The radius equals √(5) * side / 4. The quarter of the uncovered area in the middle equals (√3.2r)^2 ÷ 4 − ((90° − 2 × Atan(1 ÷ 2)) ÷ 360 × 3.2r^2 + 2(√(3.2)r ÷ 4 × √(3.2)r ÷ 2) ÷ 2)) . An overlapping area equals 2(Atan(1 ÷ 2) ÷ 360° × 3.2r^2 − (√(3.2)r ÷ 4 × √(3.2)r ÷ 2) ÷ 2) . Dividing both sides by 3.2r^2 : 1 ÷ 4−((90° − 2 × Atan(1 ÷ 2)) ÷ 360° + (1 ÷ 8)) = 2(Atan(1 ÷ 2) ÷ 360° − (1 ÷ 8) ÷ 2) . Simplifying further: 1 ÷ 4 − ((90° − 2 × Atan(1 ÷ 2)) ÷ 360°) = 2 × Atan(1 ÷ 2) ÷ 360° . Substituting 90° / 360° for 1 / 4: 90° ÷ 360° − ((90° − 2 × Atan(1 ÷ 2)) ÷ 360°)=2 × Atan(1 ÷ 2) ÷ 360° . Simplifying further: Atan(1÷2) = Atan(1÷2) . Which is equivalent to 1 = 1. 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. The area of both the square and the sum of the quadrants equals 16 right triangles with legs of a quarter, and a half of the square's sides, and its hypotenuse equal to the radius of the circle. The area of the circle equals 4 * radius / √(5))^2 = 16 / 5 × r^2 .",
 	"educationalLevel": "advanced",
 	"keywords": "radius, area",
-	  "image": [
-	"areaOfACircle.jpg",
-	"equityFigure.jpg"
-	],
+	  "image": "areaOfACircle.jpg",
      "eduQuestionType": "Area calculation",
    "usageInfo": "Logic tree position: 3b. The circle, while not a polygon, is compared to the square through geometric transformation via triangles—not as a limiting case of polygons, but as a unique comparative instance."
 	},
@@ -576,10 +573,10 @@ <h2 style="font-size:160%;margin:7px;">Exact Geometry Formulas</h2>
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-<h3 style="font-size:160%;margin:7px;">Area of a square</h3>
+<p style="font-size:160%;margin:7px;">Area of a square</p>
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 <div class="imgbox">
-    <img class="center-fit" src="square.png" alt="Square" id="square">
+    <img class="center-fit" src="square.png" alt="Square figure" id="square">
     </div>
 <br>
 <p style="margin:12px;">
@@ -611,10 +608,10 @@ <h3 style="font-size:160%;margin:7px;">Area of a square</h3>
 <br>
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-<h4 style="font-size:160%;margin:7px;">Volume of a cube</h4>
+<p style="font-size:160%;margin:7px;">Volume of a cube</p>
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 <div class="imgbox">
-    <img class="center-fit" src="cubeMarkup.jpeg" alt="Cube" id="cube">
+    <img class="center-fit" src="cubeMarkup.jpeg" alt="Cube figure" id="cube">
     </div>
 <br>
 <p style="margin:12px;">A cuboid is a 3 dimensional solid shape. Its measurable properties are width, length and height. The volume of a cuboid is a simple multiplication of the edges, width × length × height. The cubic root of the product of the edges is the edge length of the theoretical cube that has the same volume as the cuboid. 
@@ -647,10 +644,10 @@ <h4 style="font-size:160%;margin:7px;">Volume of a cube</h4>
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-<h5 style="font-size:160%;margin:7px">Trigonometry</h5>
+<p style="font-size:160%;margin:7px">Trigonometry</p>
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 <div class="imgbox">
-    <img class="center-fit" src="trigonometry.png" alt="Trigonometry" id="trigonometry">
+    <img class="center-fit" src="trigonometry.png" alt="Trigonometry figure" id="trigonometry">
     </div>
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@@ -1419,7 +1416,7 @@ <h5 style="font-size:160%;margin:7px">Trigonometry</h5>
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-<h6 style="font-size:160%;margin:7px;">Area of a regular polygon</h6>
+<p style="font-size:160%;margin:7px;">Area of a regular polygon</p>
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@@ -1948,6 +1945,103 @@ <h6 style="font-size:160%;margin:7px;">Area of a regular polygon</h6>
 </p>
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+<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mi>C</mi>
+<mo>=</mo>
+<mfrac>
+<mrow>
+<mo>(</mo>
+<mn>3.2</mn>
+<msup>
+<mi>r</mi>
+<mn>2</mn>
+</msup>
+<mo>-</mo>
+<mn>3.2</mn>
+<mo>×</mo>
+<msup>
+<mrow>
+<mo>(</mo>
+<mi>r</mi>
+<mo>-</mo>
+<mi>x</mi>
+<mo>)</mo>
+</mrow>
+<mn>2</mn>
+</msup>
+<mo>)</mo>
+</mrow>
+<mi>x</mi>
+</mfrac>
+</mrow>
+</math>
+<br>
+<br>
+<br>
+<p style="margin:12px;">
+As x is close to 0:
+</p>
+<br>
+<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
+<mrow>
+<mi>C</mi>
+<mo>=</mo>
+<mn>3.2</mn>
+<mo>×</mo>
+<mi>r</mi>
+</mrow>
+</math>
+<br>
+<br>
+<br>
+<p style="margin:12px;">
+In calculus terms:
+</p>
+<br>
+<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
+    <mrow>
+        <mi>C</mi>
+    <mo>=</mo>
+        <munder>
+      <mo>lim</mo>
+      <mrow>
+        <mi>x</mi>
+        <mo>→</mo>
+        <mn>0</mn>
+      </mrow>
+    </munder>
+        <mfrac>
+            <mrow>
+        <mn>3.2</mn> 
+        <mo>(</mo> 
+        <msup>
+          <mi>r</mi>
+          <mn>2</mn> 
+        </msup>
+        <mo>-</mo> 
+        <msup>
+          <mrow>
+            <mo>(</mo> 
+            <mi>r</mi>
+            <mo>-</mo>
+            <mi>x</mi>
+            <mo>)</mo>
+          </mrow>
+          <mn>2</mn>
+        </msup>
+        <mo>)</mo>
+      </mrow>
+      <mi>x</mi>
+    </mfrac>
+    <mo>=</mo>
+    <mn>6.4</mn>
+    <mi>r</mi>
+  </mrow>
+</math>
+</div>
+<br>
+<br>
 <br>
 <br>
 <div>