Update index.html

Author: Gmac4247

index.html

@@ -487,7 +487,7 @@ <h3 itemprop="name">Operations with Fractions</h3>
 <section itemprop="about" itemscope itemtype="http://schema.org/LearningResource" id="adding_fractions" itemref="addition">
 <h3 itemprop="name">Adding Fractions</h3>
 <br>
-<p itemprop="abstract">Adding the counters if the denominators are the same.</p>
+<p itemprop="description">Adding the counters if the denominators are the same.</p>
 <br>
 <p>Example #1:</p>
 <br>
@@ -595,7 +595,7 @@ <h3 itemprop="name">Subtracting Fractions</h3>
 <section itemprop="about" itemscope itemtype="http://schema.org/LearningResource" id="multiplying_fractions" itemref="multiplication">
 <h3 itemprop="name">Multiplying Fractions</h3>
 <br>
-<p itemprop="abstract">A fraction can be multiplied by multiplying its counter, or dividing its denominator.
+<p itemprop="description">A fraction can be multiplied by multiplying its counter, or dividing its denominator.
 <br>
 <br>
 Multiplying by a fraction means multiplying counter by counter and denominator by denominator.</p>
@@ -623,7 +623,7 @@ <h3 itemprop="name">Multiplying Fractions</h3>
 <section itemprop="about" id="dividing_fractions" itemscope itemtype="http://schema.org/LearningResource" itemref="division">
 <h3 itemprop="name">Dividing Fractions</h3>
 <br>
-<p itemprop="abstract">A fraction can be divided by dividing its counter, or multiplying its denominator.
+<p itemprop="description">A fraction can be divided by dividing its counter, or multiplying its denominator.
 <br>
 <br>
 Dividing by a fraction means multiplying by the reciprocal of that fraction.</p>
@@ -664,40 +664,37 @@ <h3 itemprop="name">Dividing Fractions</h3>
 <details>
 <summary><h3 style="margin:12px" itemprop="name">4. Powers</h3></summary>
 <br>
-<p style="margin:12px" itemprop="abstract">Raising a number or unit of measurement to a power means multiplying it by itself.</p>
+<p style="margin:12px" itemprop="description">Raising a number or unit of measurement to a power means multiplying it by itself.</p>
 <br>
-<p>Raising a value to the 3rd power means multiplying it by itself twice.
-<br>
-<br>
-Example #1: 
+<p style="margin:12px">Example #1: 
 <br>
 <br>
 2³ = 2^3 = 2 × 2 × 2 = 8
 </p>
 <br>
 <br>
-<p>Raising a value to the -4th power means the reciprocal of the 4th power. 
-<br>
+<p style="margin:12px" itemprop="description">Raising a value to a negative power means the reciprocal of the 4th power. 
+</p>
 <br>
-Example #2: 
+<p style="margin:12px">Example #2: 
 <br>
 <br>
 3^(-4) = 1 / 3⁴ = 1 / ( 3 × 3 × 3 × 3 ) = 1 / 81
 </p>
 <br>
 <br>
-<p>The result of raising a value to a fraction is a root. 
-<br>
+<p style="margin:12px" itemprop="description">The result of raising a value to a fraction is a root. 
+</p>
 <br>	
-Example #3: 
+<p style="margin:12px">Example #3: 
 <br>
 <br>
 4^(1 / 2) = √4 = 2
 </p>
 <br>
 <br>
 <br>
-<section itemprop="about" itemscope itemtype="http://schema.org/LearningResource" id="geometry"">
+<section itemprop="subjectOf" itemscope itemtype="http://schema.org/LearningResource" id="geometry"">
 <h3 style="margin:7px" itemprop="name">The 2nd and the 3rd Powers manifesting in Geometry</h3>
 <br>
 <section id="square">
@@ -808,7 +805,7 @@ <h3 style="margin:7px" itemprop="eduQuestionType">Area of a Triangle</h3>
 <br>
 The square root of half of the area of the rectangle is the side length of the theoretical square that has the same area as the triangle.</p>
 <br>
-<div itemprop="result" itemscope itemtype="https://schema.org/MathSolver"
+<div itemprop="result" itemscope itemtype="https://schema.org/MathSolver">
 <math display="block" itemprop="mathExpression" xmlns="http://www.w3.org/1998/Math/MathML">
  <mrow>
  <mi>A</mi>
@@ -831,7 +828,7 @@ <h3 style="margin:7px" itemprop="eduQuestionType">Area of a Triangle</h3>
 <p itemprop="description" style="margin:12px">The area of a triangle can also be calculated by the length of its sides.</p>
 <br>
 <div itemprop="result" itemscope itemtype="https://schema.org/MathSolver">
-<math xmlns="http://www.w3.org/1998/Math/MathML" > 
+<math <p style="margin:12px"> xmlns="http://www.w3.org/1998/Math/MathML" > 
 <mrow>
 <mi>S</mi>
     <mo>=</mo>
@@ -1001,8 +998,8 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
 <br>
 <br>
 <br>
-<div display="block" itemprop="description" id="sine">
-<math xmlns="http://www.w3.org/1998/Math/MathML">
+<div itemprop="description" id="sine">
+<math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
 <mrow>
  <mi>sine</mi>
       <mo>=</mo>
@@ -1015,8 +1012,8 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
 </div>
 <br>
 <br>
-<div display="block" itemprop="description" id="cosine">
-<math xmlns="http://www.w3.org/1998/Math/MathML">
+<div itemprop="description" id="cosine">
+<math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
 <mrow>
  <mi>cosine</mi>
       <mo>=</mo>
@@ -1029,8 +1026,8 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
 </div>
 <br>
 <br>
-<div display="block" itemprop="description" id="tangent">
-<math xmlns="http://www.w3.org/1998/Math/MathML">
+<div itemprop="description" id="tangent">
+<math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
  <mrow>
  <mi>tangent</mi>
       <mo>=</mo>
@@ -1043,8 +1040,8 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
 </div>
 <br>
 <br>
-<div display="block" itemprop="description" id="cotangent">
-<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow>
+<div itemprop="description" id="cotangent">
+<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow>
  <mi>cotangent</mi>
       <mo>=</mo>
       <mfrac>
@@ -1056,7 +1053,6 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
 </div>
 <br>
 <br>
-<br>
 <p itemprop="usageInfo" style="margin:12px">Trigonometric functions with an "Arc" pretag refer to the angle corresponding to a value of that function.</p>
 <div>
 <script>
@@ -1782,11 +1778,12 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 The arcs of the quadrants of a circumscribed circle would overlap, and intersect at the center of the square, covering it all.
 <br>
 <br>
-<strong>The arcs of the quadrants of the circle that equals in area to the square intersect right in between those limits, at the quarters on its centerlines.</strong></p>
+<strong>The arcs of the quadrants of the circle that equals in area to the square intersect right in between those limits, at the quarters on its centerlines.</strong>
 <br>
-<p itemprop="description" style="margin:12px">The ratio between the radius of the circle and the side of the square is calculable.</p>
 <br>
-<div itemprop="description" style="margin:12px">
+The ratio between the radius of the circle and the side of the square is calculable.</p>
+<br>
+<div itemprop="description">
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML">
 <mrow>
 <msup>
@@ -1905,8 +1902,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <div>
 <div id="uncovered">
-<p>Quarter of the uncovered area in the middle:
-</p>
+<p>Quarter of the uncovered area in the middle:</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -1988,8 +1984,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <br>
 <div id="overlap">
-<p>The area of an overlapping section:
-</p>
+<p>The area of an overlapping section:</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -2056,12 +2051,11 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <br>
 <div>
-<p>The equation can be simplified algebraically.
-</p>
+<p>The equation can be simplified algebraically.</p>
+<br>
 <br>
 <div>
-<p>Dividing both sides by 3.2r² :
-</p>
+<p>Dividing both sides by 3.2r² :</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -2138,8 +2132,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <br>
 <div>
-<p>Simplifying further:
-</p>
+<p>Simplifying further:</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -2195,8 +2188,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <br>
 <div>
-<p>Substituting 90° / 360° for 1 / 4 :
-</p>
+<p>Substituting 90° / 360° for 1 / 4 :</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -2258,8 +2250,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle</h3>
 <br>
 <br>
 <div>
-<p>Simplifying further:
-</p>
+<p>Simplifying further:</p>
 <br>
 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" >
 <mrow>
@@ -2381,10 +2372,9 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Circumference of a Circle</h3>
     </figure>
 <section style="margin:12px" id="pi">
 <details>
-  <summary><h4 itemprop="description" style="margin:12px">The circumference of a circle is derived from its area algebraically by subtracting a smaller circle and dividing the difference by the difference of the radii.</h4></summary>
+  <summary><h4 itemprop="description">The circumference of a circle is derived from its area algebraically by subtracting a smaller circle and dividing the difference by the difference of the radii.</h4></summary>
 <br>
-<p>
-For centuries, the circle has been a symbol of mathematical elegance—and the pi its most iconic constant. While the approximate value of 3.14159…, commonly denoted by the Greek letter pi, is widely recognized today, the historical development of this concept is less understood. Some think 'Standard Geometry' means accepting the pi. 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?
+<p>For centuries, the circle has been a symbol of mathematical elegance—and the pi its most iconic constant. While the approximate value of 3.14159…, commonly denoted by the Greek letter pi, is widely recognized today, the historical development of this concept is less understood. Some think 'Standard Geometry' means accepting the pi. 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?
 <br>
 <br>
 The constant relationship between a circle's circumference and its diameter has captivated mathematicians for millennia.
@@ -2394,8 +2384,7 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Circumference of a Circle</h3>
 The verse of 1. Kings 7:23 in the Holy Bible suggests that some estimated it as 3. 
 <br>
 <br>
-Historical records suggest that ancient Babylonians initially calculated it as 3, later they used 3.125; Egyptians estimated it as ( 16 / 9 )² ~ 3.16.
-</p>
+Historical records suggest that ancient Babylonians initially calculated it as 3, later they used 3.125; Egyptians estimated it as ( 16 / 9 )² ~ 3.16.</p>
 <br>
 <br>
 <section id="Archimedes">
@@ -2984,17 +2973,18 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Area of a Circle Segment</h3>
 <br>
     <p itemprop="description" style="margin:12px">The area of a circle segment can be calculated by subtracting a triangle from a circle slice.
 <br>
-<br>The angle of the slice is given by the ratio between the segment height and the radius of the parent circle. The base of the triangle is the chord, its height is the segment height subtracted from the radius of the parent circle. If the radius of the parent circle is unknown it can be calculated from the the length of the segment ( chord ).</p>
+<br>
+The angle of the slice is given by the ratio between the segment height and the radius of the parent circle. The base of the triangle is the length of the segment. That is called a chord. The height of the triangle is the segment height subtracted from the radius of the parent circle. If the radius of the parent circle is unknown it can be calculated from the chord.</p>
 <br>
 <math style="margin:12px" xmlns="http://www.w3.org/1998/Math/MathML">
   <mi>radius</mi>
   <mo>=</mo>
   <mfrac>
     <mrow>
-      <msup><mi>chord</mi><mn>2</mn></msup>
+      <msup><mi>l</mi><mn>2</mn></msup>
       <mo>+</mo>
       <mn>4</mn>
-      <msup><mi>height</mi><mn>2</mn></msup>
+      <msup><mi>h</mi><mn>2</mn></msup>
     </mrow>
     <mrow>
       <mn>8</mn>
@@ -4740,7 +4730,6 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Volume of a Tetrahedron</h3>
 This is the only exact, self-contained geometric framework grounded in the first principles of mathematics, providing exact formulas for real-world applications.</strong></p>
 <p itemprop="disambiguatingDescription" 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>
-<br>
 <p itemprop="usageInfo" style="margin:12px">
 This is the best framework for exact geometric calculations in engineering design solutions, computer graphics rendering, algorithm optimization, and navigation.</p>
 <br>
@@ -4758,14 +4747,13 @@ <h3 itemprop="eduQuestionType" style="margin:7px">Volume of a Tetrahedron</h3>
 <br>
 Algebraic Manipulation
 <br>
-Simplifying equations to ensure consistency and precision.
-</p>
+Simplifying equations to ensure consistency and precision.</p>
+<br>
+<br>
 <p itemprop="copyrightNotice" style="margin:12px">
 ® All rights reserved 2025 Gaál Sándor</p>
 </div>
 <br>
-<br> 
-<br>
 <footer>
 <a style="margin:9px" itemscope itemtype="http://schema.org/AboutPage" href="about">About</a>
 <br>