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‎index.html‎

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"name" :"Szentendre, Hungary"},
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"mainEntity" : {
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"@type": ["MathSolver", "LearningResource"],
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"name" : "Exact Geometry",
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"name" : "Core Geometric System ™",
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"inLanguage":"en-us",
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"url": "https://basic-geometry.github.io/",
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"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.",
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"potentialAction": [
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{
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{
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"@type": "SolveMathAction",
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"name": "Area of a square",
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"description": "The basis of area calculation",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "Side length of the square",
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"mathExpression-output": "Area of the square",
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"mathExpression-input": "a=5 A=?",
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"mathExpression-output": "Math.pow(side, 2)",
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"image": "square.png",
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"eduQuestionType": "Area calculation",
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"name": "Calculating the area of a square",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "Math.pow(side, 2)"
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}
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"eduQuestionType": "Area calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a cube",
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"description": "The basis of volume calculation",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "Edge length of the cube",
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"mathExpression-output": "Volume of the cube",
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"mathExpression-input": "a=5 V=?",
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"mathExpression-output": "Math.pow(edge, 3)",
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"image": "cubeMarkup.jpeg",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a cube",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "Math.pow(edge, 3)"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Area of a circle",
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"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.",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius of the circle, and optionally the angle of rotation in case of a circle slice",
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"mathExpression-output":
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"The area of the circle",
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"mathExpression-input": "r=5 A=?",
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"mathExpression-output": "angle of rotation / 360 * 3.2 * Math.pow(radius, 2)",
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"image": "areaOfACircle.jpg",
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"eduQuestionType": "Area calculation",
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"name": "Calculating the area of a circle",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2)"
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}
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},
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"eduQuestionType": "Area calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Area of a circle segment",
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"description": "Calculates the area of a circle segment based on its radius and the height of the segment",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius of the circle, optionally the angle of rotation in case of a section",
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"mathExpression-output": "The circumference of the circle",
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"image": "circumference.jpg",
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"eduQuestionType": "Length calculation",
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"name": "Calculating the circumference of a circle",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "angleOfRotation / 360 * 6.4 * radius"
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}
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"mathExpression-input": "r=5 h=2 A=?",
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"mathExpression-output": "Math.acos((radius-segmentHeight)/radius)*Math.pow(radius, 2)-Math.sin(Math.acos((radius-segmentHeight)/radius))*(radius-segmentHeight)*radius",
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"image": "circleSegment.jpg",
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"eduQuestionType": "Area calculation"
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},
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{
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"@type": "SolveMathAction",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The height of the segment, and the radius of the full circle",
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"mathExpression-output": "The area of the circle segment",
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"image": "circleSegment.jpg",
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"eduQuestionType": "Area calculation",
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"name": "Calculating the area of a circle segment",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "Math.acos((radius-segmentHeight)/radius)*Math.pow(radius, 2)-Math.sin(Math.acos((radius-segmentHeight)/radius))*(radius-segmentHeight)*radius"
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}
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},
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"name": "Circumference of a circle",
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"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.",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "r=5 C=?",
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"mathExpression-output": "angle of rotation / 360 * 6.4 * radius",
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"image": "circumference.jpg",
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"eduQuestionType": "Length calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a sphere",
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"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.",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius of the sphere, and optionally the angle of rotation in case of a sphere slice",
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"mathExpression-output": "The volume of the sphere",
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"mathExpression-input": "r=3 V=?",
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"mathExpression-output": "angle of rotation / 360 * Math.pow((Math.sqrt(3.2) * radius), 3)",
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"image": "sphereAndCubeMarkup.jpeg",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a sphere",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "angleOfRotation/360*(3.2^(1/2)*radius)^3"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a spherical cap",
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"description": "Calculates the volume of a spherical cap based on the radius of the sphere and the cap.",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius of the bottom of the cap, and the full sphere",
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"mathExpression-output": "The volume of the spherical cap",
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"mathExpression-input": "r1=5 r2=3 V=?",
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"mathExpression-output": "1.6 * Math.pow(sphereSliceBottomRadius, 2) * Math.sqrt(3.2) * (1 - Math.sin(Math.acos(sphereSliceBottomRadius / sphereRadius)))",
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"image": "sphericalCap.jpg",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a spherical cap",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "1.6 * Math.pow(sphereSliceBottomRadius, 2) * Math.sqrt(3.2) * (1 - Math.sin(Math.acos(sphereSliceBottomRadius / sphereRadius)))"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius and the height of the cylinder",
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"mathExpression-output": "The volume of the cylinder",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a cylinder",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2) * height"
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}
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},
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{
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"@type": "SolveMathAction",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The radius and the height of the cylinder",
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"mathExpression-output": "The surface of the cylinder",
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"eduQuestionType": "Area calculation",
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"name": "Calculating the surface of a cylinder",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "6.4 * radius * height"
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}
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},
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{
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"@type": "SolveMathAction",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The bottom radius and the height of the cone",
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"mathExpression-output": "The volume of the cone",
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"name": "Volume of a cone",
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"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.",
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"mathExpression-input": "r=5 H=3 V=?",
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"mathExpression-output": "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2) * height / Math.sqrt(8)",
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"image": [
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"coneAndSphereMarkup.jpeg",
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"sphereAndConeMarkup.jpeg",
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"octantSphereQuarterCone.jpeg",
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"coneAndSphereComparison.png"
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],
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a cone",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "angleOfRotation / 360 * 3.2 * Math.pow(radius, 2) * height / Math.sqrt(8)"
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}
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},
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a frustum cone",
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"description": "Calculates the exact volume of a frustum cone based on its top and bottom diameter and height",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The top and the bottom diameter and the height of the cone",
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"mathExpression-output": "The volume of the frustum cone",
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"mathExpression-input": "d1=5 d2=2 h=3 V=?",
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"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)",
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"image": "frustumOfConeMarkup.png",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a frustum cone",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "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)"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"name":"Surface area of a cone",
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"description": "Calculates the exact surface area of a cone based on its radius and height",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The bottom radius and the height of the cone",
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"mathExpression-output": "The surface area of the cone",
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"mathExpression-input": "r=5 H=3 A=?",
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"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))))",
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"image": "coneMarkup.jpeg",
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"eduQuestionType": "Area calculation",
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"name":"Calculating the surface of a cone",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "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))))"
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}
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"eduQuestionType": "Area calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a pyramid",
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"description": "Calculates the exact volume of a pyramid based on the bottom area and the height of the pyramid",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The bottom area and the height of the pyramid",
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"mathExpression-output": "The volume of the pyramid",
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"mathExpression-input": "A=5 H=3 V=?",
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"mathExpression-output": "baseArea * height / Math.sqrt(8)",
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"image": [
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"conePyramidVolumeMarkup.jpeg",
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"tetraFrame.jpeg"
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],
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a pyramid",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "baseArea * height / Math.sqrt(8)"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The top and the bottom edge and the height of the frustum pyramid",
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"mathExpression-output": "The volume of the frustum pyramid",
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"name": "Volume of a frustum pyramid",
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"description": "Calculates the exact volume of a frustum pyramid based on its top and bottom edge length and height",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "a=5 b=3 H=2 V=?",
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"mathExpression-output": "frustumHeight * (Math.pow(bottomEdge, 2) * (1 / (1 - topEdge / bottomEdge)) - Math.pow(topEdge, 2) * (1 / (1 - topEdge / bottomEdge) - 1)) / Math.sqrt(8)",
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"image": "frustumOfPyramidMarkup.png",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a frustum pyramid",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "frustumHeight * (Math.pow(bottomEdge, 2) * (1 / (1 - topEdge / bottomEdge)) - Math.pow(topEdge, 2) * (1 / (1 - topEdge / bottomEdge) - 1)) / Math.sqrt(8)"
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}
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"eduQuestionType": "Volume calculation"
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},
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{
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"@type": "SolveMathAction",
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"name": "Volume of a tetrahedron",
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"description": "Calculates the exact volume of a tetrahedron based on its edge length",
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"target": "https://basic-geometry.github.io/",
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"mathExpression-input": "The edge length of the tetrahedron",
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"mathExpression-output": "The volume of the tetrahedron",
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"mathExpression-input": "a=5 V=?",
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"mathExpression-output": "Math.pow(edge, 3) / 8",
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"image": "tetrahedronMarkup.jpeg",
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"eduQuestionType": "Volume calculation",
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"name": "Calculating the volume of a tetrahedron",
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"additionalProperty": {
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"@type": "PropertyValue",
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"name": "Formula",
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"value": "Math.pow(edge, 3) / 8"
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}
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"eduQuestionType": "Volume calculation"
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}
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]
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},
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"mainEntityOfPage": {

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