Update index.html

Markup

Author: Gmac4247

index.html

@@ -179,9 +179,9 @@
 },
 "dateCreated" :"2019-01-11",
 "datePublished":"2020-01-11",
-"dateModified" :"2025-05-24",
-"description" :"Introducing the best-established and most accurate framework to calculate area and volume.",
-"disambiguatingDescription": "Providing precise formulas based on empirical considerations and fundamental relationships instead of the conventional approximations.",
+"dateModified" :"2025-05-27",
+"description" : "Introducing the best-established and most accurate framework to calculate area and volume.",
+"disambiguatingDescription": "Exact, empirically grounded and rigorously proven formulas over the conventional approximations.",
 "headline":"Exact geometry",
 "inLanguage":"en-us",
 "image":[
@@ -197,6 +197,9 @@
  "mainEntity" : {
 "@type": ["MathSolver", "LearningResource"],
  "name" : "Core Geometric System ™",
+"description":"The best-established, empirically grounded, rigorously proven and most accurate framework."
+ "disambiguatingDescription": "Exact formulas over the traditional approximations. More accurate for real-world applications.",
+"keywords":"Exact Formulas, Area of a Circle, Circumference of a Circle, Volume of a Sphere, Volume of a Cone, Volume of a Pyramid, Volume of a Tetrahedron, Volume of a Frustum", 
  "inLanguage":"en-us",
 "url": "https://basic-geometry.github.io/", 
 "usageInfo":"Calculate area and volume with enhanced accuracy using the Core Geometric System ™. This innovative framework provides a practical alternative to traditional methods, rooted in comparative geometry and scaling principles. Learn exact formulas for circles (A = 3.2r^2, C = 6.4r), spheres (V = (√(3.2)r)^3), cones, and more, with applications in engineering, computer graphics, and scientific research.",
@@ -227,8 +230,10 @@
 {
      "@type": "SolveMathAction",
 	"name": "Area of a circle",
-"description": "Calculates the exact area of a circle using a refined approach that replaces traditional π-based approximations. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
-     "target": "https://basic-geometry.github.io/",
+""description": "The exact area of a circle based on direct comparison with a square.",
+     "disambiguatingDescription": "Replaces traditional π-based approximations ensuring greater accuracy in real-world measurements.",
+	"keywords":"Exact Area of a Circle, Direct Comparison, Quadrants Method, Rigorously Proven",
+	 "target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r=5 A=?",
      "mathExpression-output": "angle of rotation / 360 * 3.2 * Math.pow(radius, 2)",
      "image": "areaOfACircle.jpg",
@@ -238,8 +243,10 @@
 {
      "@type": "SolveMathAction",
        "name":  "Area of a circle segment",
-       "description":  "Calculates the area of a circle segment based on its radius and the height of the segment",
-     "target": "https://basic-geometry.github.io/",
+"description": "The exact area of a circle segment by subtracting a triangle from a circle slice.",
+    "disambiguatingDescription": "Equivalent to the conventional method, but relies on trigonometric functions.",
+	"keywords":"Exact Area of a Circle Segment, Rigorously Proven",
+	"target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r=5 h=2 A=?",
      "mathExpression-output": "Math.acos((radius-segmentHeight)/radius)*Math.pow(radius, 2)-Math.sin(Math.acos((radius-segmentHeight)/radius))*(radius-segmentHeight)*radius", 
      "image": "circleSegment.jpg",
@@ -249,7 +256,9 @@
 {
      "@type": "SolveMathAction",
 "name": "Circumference of a circle",
-     "description": "Calculates the exact circumference of a circle using a refined approach that replaces traditional π-based approximations. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
+"description": "The exact circumference of a circle using a refined approach that replaces traditional π-based approximations. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
+"disambiguatingDescription": "Replaces traditional π-based approximations ensuring greater accuracy in real-world measurements.",
+	"keywords":"Circumference of a Circle, Algebraic Derivation from the Area, Rigorously Proven",
 	"target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r=5 C=?", 
      "mathExpression-output": "angle of rotation / 360 * 6.4 * radius",
@@ -260,8 +269,10 @@
 {
      "@type": "SolveMathAction",
 	"name": "Volume of a sphere",
-	 "description": "Calculates the exact volume of a sphere by directly comparing it to a cube. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
-     "target": "https://basic-geometry.github.io/",
+	 "description": "The exact volume of a sphere by directly comparing it to a cube. Direct shape relationships ensure greater accuracy in real-world measurements.",
+    "disambiguatingDescription": "More accurate than the traditional exhaustion method based formula which is a very rough underestimate.
+	"keywords" : "Direct Comparison, Exact Volume, Rigorously proven",
+  "target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r=3 V=?",
      "mathExpression-output": "angle of rotation / 360 * Math.pow((Math.sqrt(3.2) * radius), 3)", 
      "image": "sphereAndCubeMarkup.jpeg",
@@ -271,8 +282,10 @@
 {
      "@type": "SolveMathAction",
          "name": "Volume of a spherical cap",
-        "description": "Calculates the volume of a spherical cap based on the radius of the sphere and the cap.",
-     "target": "https://basic-geometry.github.io/",
+        "description": "The volume of a spherical cap based on the radius of the sphere and the cap.",
+     "disambiguatingDescription": "More accurate than the conventional formula ",
+	"keywords":"Spherical cap volume",
+	"target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r1=5 r2=3 V=?",
      "mathExpression-output": "1.6 * Math.pow(sphereSliceBottomRadius, 2) * Math.sqrt(3.2) * (1 - Math.sin(Math.acos(sphereSliceBottomRadius / sphereRadius)))", 
      "image": "sphericalCap.jpg",
@@ -284,7 +297,9 @@
      "@type": "SolveMathAction",
      "target": "https://basic-geometry.github.io/",
      "name": "Volume of a cone",
-	 "description": "Calculates the exact volume of a cone by comparing the volume of a quarter cone to an octant of a sphere with an equal radius. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
+	 "description": "The exact volume of a cone by comparing the volume of a quarter cone to an octant sphere with an equal radius.",
+	"disambiguatingDescription": "Instead of the inaccurate base×height/3 approximation, direct shape relationships ensure greater accuracy in real-world measurements.
+	"keywords":"Volume of a cone, Octant sphere comparison",
 	"mathExpression-input": "r=5 H=3 V=?",
      "mathExpression-output": "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2) * height / Math.sqrt(8)",
      "image": [
@@ -299,8 +314,10 @@
 {
      "@type": "SolveMathAction",
        "name": "Volume of a frustum cone",
-       "description": "Calculates the exact volume of a frustum cone based on its top and bottom diameter and height",
-     "target": "https://basic-geometry.github.io/",
+              "description": "The exact volume of a frustum cone based on its top and bottom diameter and height",
+     "disambiguatingDescription": "The formula subtracts the missing tip from a theoretical full cone",
+	"keywords": "Frustum cone volume, universal subtraction method",
+  "target": "https://basic-geometry.github.io/",
      "mathExpression-input": "d1=5 d2=2 h=3 V=?",
      "mathExpression-output": "frustumHeight * (4 / 5 * Math.pow(bottomDiameter, 2) * (1 / (1 - topDiameter / bottomDiameter)) - 4 / 5 * Math.pow(topDiameter, 2) * (1 / (1 - topDiameter / bottomDiameter) - 1)) / Math.sqrt(8)",
      "image": "frustumOfConeMarkup.png",
@@ -310,8 +327,10 @@
 {
      "@type": "SolveMathAction",
        "name":"Surface area of a cone",
-       "description": "Calculates the exact surface area of a cone based on its radius and height",
-     "target": "https://basic-geometry.github.io/",
+       "description": "The exact surface area of a cone based on its radius and height.",
+     "disambiguatingDescription": "Equivalent to the conventional formula, but relies on the real height."
+	"keywords": "Surface area of a cone based on its radius and height.",
+	 "target": "https://basic-geometry.github.io/",
      "mathExpression-input": "r=5 H=3 A=?",
      "mathExpression-output": "3.2 * (Math.pow(radius, 2) + (Math.pow(radius, 2) + Math.pow(height, 2)) * (radius / Math.sqrt(Math.pow(radius, 2) + Math.pow(height, 2))))", 
      "image": "coneMarkup.jpeg",
@@ -321,8 +340,10 @@
  {
      "@type": "SolveMathAction",
        "name": "Volume of a pyramid",
-     "description": "Calculates the exact volume of a pyramid based on the bottom area and the height of the pyramid",
-     "target": "https://basic-geometry.github.io/",
+     "description": "The exact volume of a pyramid based on its bottom area and height using the coefficient of the volume of a cone",
+     "disambiguatingDescription": "Instead of the inaccurate base×height/3 approximation, direct shape relationships ensure greater accuracy in real-world measurements.
+	"keywords":"Pyramid volume, rigorously proven",
+	 "target": "https://basic-geometry.github.io/",
      "mathExpression-input": "A=5 H=3 V=?",
      "mathExpression-output": "baseArea * height / Math.sqrt(8)", 
      "image": [
@@ -335,8 +356,10 @@
 {
      "@type": "SolveMathAction",
         "name": "Volume of a frustum pyramid",
-    "description": "Calculates the exact volume of a frustum pyramid based on its top and bottom edge length and height",
-       "target": "https://basic-geometry.github.io/",
+    "description": "The exact volume of a frustum pyramid based on its top and bottom base area and height",
+     "disambiguatingDescription": "The formula subtracts the missing tip from a theoretical full pyramid. Universally applicable",
+	"keywords": "Frustum pyramid volume, universal subtraction method, rigorously proven",
+	"target": "https://basic-geometry.github.io/",
      "mathExpression-input": "a=5 b=3 H=2 V=?",
        "mathExpression-output": "frustumHeight * (Math.pow(bottomEdge, 2) * (1 / (1 - topEdge / bottomEdge)) - Math.pow(topEdge, 2) * (1 / (1 - topEdge / bottomEdge) - 1)) / Math.sqrt(8)",
      "image": "frustumOfPyramidMarkup.png",
@@ -346,8 +369,10 @@
 {  
      "@type": "SolveMathAction",
        "name": "Volume of a tetrahedron",
-       "description": "Calculates the exact volume of a tetrahedron based on its edge length",
-     "target": "https://basic-geometry.github.io/",
+       "description": "The exact volume of a tetrahedron based on its edge length",
+     "disambiguatingDescription": "Based on the base×height/√8 formula, instead of the inaccurate conventional coefficient",
+"keywords": "tetrahedron volume, rigorously proven",
+	"target": "https://basic-geometry.github.io/",
      "mathExpression-input": "a=5 V=?",
      "mathExpression-output": "Math.pow(edge, 3) / 8", 
      "image": "tetrahedronMarkup.jpeg",