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<!DOCTYPE html>
<html lang='en'>
<head>
<meta charset="UTF-8">
<title>Home of Basic Geometry</title>
<link rel="alternate" type="text/html" href="https://cgs-mcp-server-authless.gmac4247-ac0.workers.dev" title="Core Geometric System MCP Server Home">
<link rel="alternate" type="application/mcp+json" href="https://cgs-mcp-server-authless.gmac4247-ac0.workers.dev/manifest.json" title="Basic Geometry MCP Server Manifest">
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<meta name="author" content="Gaál Sándor">
<meta name="description" content="Universally applicable self-consistent geometric framework of physically grounded exact area and volume calculation formulas for regular shapes.">
<meta name="keywords" content="Core Geometric System, Exact Geometric Calculations, Analysis, Engineering Design Solutions, Computer Graphics Rendering, Algorithm Optimization, Navigation">
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<style>
* {
margin: 0;
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}
.imgbox {
display: grid;
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}
.center-fit {
max-width: 100%;
max-height: 100vh;
margin: auto;
}
#cookie-notice {
position: fixed;
top: 0;
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background-color: #f2f2f2;
padding: 10px;
display: flex;
align-items: center;
justify-content: space-between;
box-shadow: 0px 0px 10px rgba(0, 0, 0, 0.2);
z-index: 1;
}
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margin-right: 20px;
}
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display: flex;
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}
#cookie-notice-buttons button {
margin-left: 10px;
background-color: #4CAF50;
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padding: 10px;
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display: inline-block;
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</style>
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<body>
<div itemscope itemtype="https://schema.org/Statement" id="cookie-notice">
<div id="cookie-notice-text">
Welcome!
This website uses cookies to ensure you get the best experience on it.
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</div>
<main itemscope itemtype="https://schema.org/LearningResource https://schema.org/MathSolver" id="cgs">
<meta itemprop="name" content="Core Geometric System ™">
<meta itemprop="accessMode" content="mathOnVisual">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Equations with figures and explanations">
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact area of a circle">
<meta itemprop="description" content="Area = 3.2 × radius^2, derived from direct circle to square comparison">
<meta itemprop="disambiguatingDescription" content="Not the abstract pi~3.14 approximation">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact circumference of a circle">
<meta itemprop="description" content="Circumference = 6.4 × radius, derived from the exact area of a circle by subtracting a circle from a larger one and dividing the difference by the theoretical width of the circumference.">
<meta itemprop="disambiguatingDescription" content="Not the abstract pi~3.14 approximate">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
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<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact volume of a sphere">
<meta itemprop="description" content="Volume = ( 4 × radius / 5^(1/2) )^3, derived from direct sphere to cube comparison.">
<meta itemprop="disambiguatingDescription" content="Not the abstract 4×radius³/3×3.14... approximate">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact volume of a cone">
<meta itemprop="description" content="Volume = 3.2 × radius^2 × height / 8^(1/2), derived from comparing a quadrant cone with height equal to radius to an octant sphere with equal radius.">
<meta itemprop="disambiguatingDescription" content="Not the abstract radius^2×height/3×3.14… approximate">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact surface area of a cone">
<meta itemprop="description" content="Surface area = 3.2 × radius × (radius + sqrt(radius^2 + height^2)), calculated as a circle slice.">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact area of a triangle">
<meta itemprop="description" content="Surface area = Area = base × height / 2">
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</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact area of a regular polygon">
<meta itemprop="description" content="Surface area = Area = sideCount / 4 × ctg(180° / sideCount) × sideLength^2">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Area of a circle segment">
<meta itemprop="description" content="Area = acos((radius − height) / radius) × radius^2 − sin(acos((radius − height) / radius)) × radius × (radius − height)">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
</div>
<div itemprop="assesses" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Exact volume of a pyramid">
<meta itemprop="description" content="Volume=base×height/8^(1/2), calculated as a cone with a polygonal base.">
<meta itemprop="disambiguatingDescription" content="Not the abstract base×height/3 approximate">
<meta itemprop="inDefinedTermSet" content="Core Geometric System ™">
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<meta itemprop="educationalRole" content="For teachers and learners">
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<meta itemprop="familyName" content="Gaál">
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<meta itemprop="educationalLevel" content="from basic to advanced">
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<h1 itemprop="headline" style="font-size:130%;margin:7px">Featuring The Core Geometric System ™</h1>
<br><br>
<section itemprop="mathExpression" itemscope itemtype="https://schema.org/SolveMathAction" id="circle">
<meta itemprop="eduQuestionType" content="Circle area calculation">
<div itemprop="object" itemscope itemtype="https://schema.org/QuantitativeValue">
<meta itemprop="name" content="radius">
<meta itemprop="minValue" content="0">
</div>
<meta itemprop="target" content="https://basic-geometry.github.io?q={circle_radius=1_area=?}">
<div itemprop="actionProcess" itemscope itemtype="https://schema.org/HowTo">
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</div>
<div itemprop="tool" itemscope itemtype="https://schema.org/HowToTool">
<div itemprop="item" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="Exact circle area formula">
<meta itemprop="name" content="Area of a circle">
<meta itemprop="value" content="3.2 × radius^2">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<h2 itemprop="name" style="margin:7px">Area of a Circle</h2>
<br>
<figure itemprop="image" class="imgbox" itemscope itemtype="https://schema.org/ImageObject">
<img class="center-fit" src="circle.png" alt="Circle area">
<div itemprop="creator" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<div itemprop="copyrightHolder" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
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<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
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</div>
<meta itemprop="url" content="LICENSE.txt">
</div>
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/circle.png">
</figure>
<br>
<p style="margin:12px" itemprop="description"><strong>The area of a circle is defined by comparing it to a square since that is the base of area calculation.</strong></p>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="1">
<p itemprop="abstract">The circle can be cut into four quadrants, each placed with their origin on the vertices of a square.
<br><br>
In this layout the arcs of the quadrants of an inscribed circle would meet at the midpoints of the sides of the square, leaving some of the square uncovered.
<br><br>
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>
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="2">
<p itemprop="description">The length of the centerline equals the side length of the square.
<br>
The quarter of the centerline and the half of the side of the square form two legs of a right triangle with its hypotenuse equal to the radius.
<br>
The ratio between the radius and the side length is calculable.</p>
<br>
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="The ratio between the squared radius of the circle and an equiareal square">
<span itemprop="name">radius² </span>=
<meta itemprop="value" content="( side / 4 )^2 + ( side / 2 )^2">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>side</mi>
<mn>4</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>side</mi>
<mn>2</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math>
</div>
</div>
<br>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="3">
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="The ratio between the radius of the circle and an equiareal square">
<span itemprop="name">radius </span>=
<meta itemprop="value" content="side × 5^(1/2) / 4">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>side</mi>
<mn>4</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mfrac>
<mi>side</mi>
<mn>4</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow>
<msqrt>
<mrow>
<mn>5</mn>
<mo>×</mo>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mi>side</mi>
<mn>4</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
<mo>×</mo>
<mfrac>
<mi>side</mi>
<mn>4</mn>
</mfrac>
</mrow>
</math>
</div>
</div>
</div>
<br>
<div style="margin:12px" itemprop="result" itemscope itemtype="https://schema.org/PropertyValue">
<p itemprop="description">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.</p>
<br><br>
<meta itemprop="description" content="Exact circle area formula">
<span itemprop="name">A<sub>(circle)</sub> </span>=
<meta itemprop="value" content="3.2 × radius^2">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow>
<mfrac>
<mn>16</mn>
<mn>5</mn>
</mfrac>
<mo>×</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</math>
</div>
<br><br>
<div style="margin:12px" id="circle_area_calculator" itemprop="subjectOf" itemscope itemtype="https://schema.org/SoftwareApplication">
<meta itemprop="name" content="Circle area calculator">
<meta itemprop="accessMode" content="mathOnVisual">
<meta itemprop="accessibilityControl" content="fullKeyboardControl">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Enter numbers and read result">
<meta itemprop="applicationCategory" content="Geometric Calculator">
<div itemprop="creator" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<div itemprop="copyrightHolder" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<meta itemprop="copyrightNotice" content="® All rights reserved">
<meta itemprop="copyrightYear" content="2026">
<div itemprop="license" itemscope itemtype="https://schema.org/DigitalDocument">
<meta itemprop="name" content="Shared Software License Agreement">
<div itemprop="author" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
<meta itemprop="url" content="https://x.com/gmac4247">
</div>
<meta itemprop="description" content="® All rights reserved.">
<div itemprop="hasDigitalDocumentPermission" itemscope itemtype="https://schema.org/DigitalDocumentPermission">
<meta itemprop="permissionType" content="https://schema.org/ReadPermission">
<div itemprop="grantee" itemscope itemtype="https://schema.org/Audience">
<meta itemprop="audienceType" content="public">
</div>
</div>
<meta itemprop="url" content="LICENSE.txt">
</div>
<meta itemprop="description" content="Calculates the area of a circle as 3.2×radius^2.">
<meta itemprop="disambiguatingDescription" content="Does not calculate the area as radius^2×3.14...">
<meta itemprop="featureList" content="Calculates the area of a circle from its radius with an exact formula.">
<meta itemprop="inLanguage" content="en">
<meta itemprop="interactivityType" content="Active">
<meta itemprop="isAccessibleForFree" content="true">
<meta itemprop="isFamilyFriendly" content="true">
<meta itemprop="operatingSystem" content="Web">
<meta itemprop="url" content="https://basic-geometry.github.io#circle_area_calculator">
<meta itemprop="usageInfo" content="Enter the radius">
<label for="circle-radius-a">Radius:</label>
<input id="circle-radius-a" type="number" value="1" step="any">
<br><br>
<p id="circle-area"></p>
<script type="module" async>
function circleArea(radius) {
return 3.2 * radius * radius;
}
document.getElementById('circle-radius-a').addEventListener('input', function () {
const radius = parseFloat(this.value);
if (isNaN(radius)) {
document.getElementById('circle-area').innerText = '';
return;
}
document.getElementById('circle-area').innerText =
`Area = ${circleArea(radius).toFixed(5)} square units`;
});
</script>
</div>
</section>
<br><br><br>
<section itemprop="mathExpression" itemscope itemtype="https://schema.org/SolveMathAction" id="circumference">
<meta itemprop="eduQuestionType" content="Circumference calculation">
<div itemprop="object" itemscope itemtype="https://schema.org/QuantitativeValue">
<meta itemprop="name" content="radius">
<meta itemprop="minValue" content="0">
</div>
<meta itemprop="target" content="https://basic-geometry.github.io?q={circle_radius=1_circumference=?}">
<div itemprop="actionProcess" itemscope itemtype="https://schema.org/HowTo">
<div itemprop="estimatedCost" itemscope itemtype="https://schema.org/MonetaryAmount">
<meta itemprop="currency" content="USD">
<meta itemprop="value" content="0">
</div>
<div itemprop="tool" itemscope itemtype="https://schema.org/HowToTool">
<div itemprop="item" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="Exact circle circumference formula">
<meta itemprop="name" content="Circumference of a circle">
<meta itemprop="value" content="6.4 × radius">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<h3 itemprop="name" style="margin:7px">Circumference of a Circle</h3>
<br>
<figure itemprop="image" class="imgbox" itemscope itemtype="https://schema.org/ImageObject">
<img class="center-fit" src="circumference.png" alt="The circumference of a circle is derived from its area algebraically by subtracting a smaller circle">
<div itemprop="creator" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<div itemprop="copyrightHolder" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<meta itemprop="copyrightNotice" content="® All rights reserved">
<meta itemprop="copyrightYear" content="2026">
<div itemprop="license" itemscope itemtype="https://schema.org/DigitalDocument">
<meta itemprop="name" content="Shared Software License Agreement">
<div itemprop="author" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
<meta itemprop="url" content="https://x.com/gmac4247">
</div>
<meta itemprop="description" content="® All rights reserved.">
<div itemprop="hasDigitalDocumentPermission" itemscope itemtype="https://schema.org/DigitalDocumentPermission">
<meta itemprop="permissionType" content="https://schema.org/ReadPermission">
<div itemprop="grantee" itemscope itemtype="https://schema.org/Audience">
<meta itemprop="audienceType" content="public">
</div>
</div>
<meta itemprop="url" content="LICENSE.txt">
</div>
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/circumference.png">
</figure>
<br>
<p style="margin:12px" itemprop="description"><strong>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.</strong></p>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="1">
<p itemprop="abstract">Let x represent the theoretical width of the circumference, which is a very small number.
<br><br>
The difference between the shape of the straightened circumference and a quadrilateral is negligible.
<br><br>
The length of the two shorter sides of the quadrilateral is x.
<br><br>
The length of the two longer sides is the area of the resulting ring divided by x.</p>
<br>
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="Circumference derivation">
<span itemprop="name">Circumference </span>=
<meta itemprop="value" content="( 3.2 × radius^2 - 3.2 × ( radius - circumferenceWidth )^2 ) / circumferenceWidth">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mfrac>
<mrow>
<mo>(</mo>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<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>
</div>
</div>
<br>
<details style="margin:12px">
<summary><h4>Expand</h4></summary>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="2">
<div>
<p itemprop="description">the term (r - x)²:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>-</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>=</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
<mo>⁢</mo> <!-- invisible times -->
<mi>x</mi>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math>
</div>
</div>
<br><br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="3">
<div>
<p itemprop="description">Substitute this back into the original expression:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mfrac>
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<mo>(</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>2</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
<mo>⁢</mo> <!-- invisible times -->
<mi>x</mi>
<mo>+</mo>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</math>
</div>
</div>
<br><br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="4">
<div>
<p itemprop="description">Distribute the 3.2 inside the parentheses:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mfrac>
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>6.4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
<mi>x</mi>
<mo>-</mo>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</math>
</div>
</div>
<br><br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="5">
<div>
<p itemprop="description">Simplify the numerator:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mfrac>
<mrow>
<mn>6.4</mn>
<mi>r</mi>
<mi>x</mi>
<mo>-</mo>
<mn>3.2</mn>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</math>
</div>
</div>
<br><br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="6">
<div>
<p itemprop="description">Factor out x from the numerator:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>(</mo>
<mn>6.4</mn>
<mi>r</mi>
<mo>-</mo>
<mn>3.2</mn>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
</math>
</div>
</div>
<br><br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="7">
<div>
<p itemprop="description">Cancel out the x in the numerator and denominator:</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mn>6.4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
<mo>-</mo>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>x</mi>
</mrow>
</math>
</div>
</div><br><br><br>
<div style="margin:12px">
<p>In calculus terms:</p>
<br><br>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>C</mi>
<mo>=</mo>
<mrow>
<munder>
<mo>lim</mo>
<mrow>
<mi>x</mi>
<mo>→</mo>
<mn>0</mn>
</mrow>
</munder>
<mfrac>
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<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>
</mrow>
<mo>=</mo>
<mrow>
<mn>6.4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
</mrow>
</math>
</div>
<br><br>
</details>
</div>
<br>
<div style="margin:12px" itemprop="result" itemscope itemtype="https://schema.org/PropertyValue">
<p itemprop="description">The length of the circumference approaches 6.4 × radius as its thickness approaches 0.</p>
<br><br>
<meta itemprop="description" content="Exact circumference formula">
<span itemprop="name">Circumference </span>=
<meta itemprop="value" content="6.4 × radius">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mn>6.4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>radius</mi>
</mrow>
</math>
</div>
<br><br>
<div style="margin:12px" id="circumference_calculator" itemprop="subjectOf" itemscope itemtype="https://schema.org/SoftwareApplication">
<meta itemprop="name" content="Circumference calculator">
<meta itemprop="accessMode" content="mathOnVisual">
<meta itemprop="accessibilityControl" content="fullKeyboardControl">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Enter numbers and read results">
<meta itemprop="applicationCategory" content="Geometric Calculator">
<div itemprop="creator" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<div itemprop="copyrightHolder" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<meta itemprop="copyrightNotice" content="® All rights reserved">
<meta itemprop="copyrightYear" content="2026">
<div itemprop="license" itemscope itemtype="https://schema.org/DigitalDocument">
<meta itemprop="name" content="Shared Software License Agreement">
<div itemprop="author" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
<meta itemprop="url" content="https://x.com/gmac4247">
</div>
<meta itemprop="description" content="® All rights reserved.">
<div itemprop="hasDigitalDocumentPermission" itemscope itemtype="https://schema.org/DigitalDocumentPermission">
<meta itemprop="permissionType" content="https://schema.org/ReadPermission">
<div itemprop="grantee" itemscope itemtype="https://schema.org/Audience">
<meta itemprop="audienceType" content="public">
</div>
</div>
<meta itemprop="url" content="LICENSE.txt">
</div>
<meta itemprop="description" content="Calculates the circumference as 3.2×radius^2.">
<meta itemprop="disambiguatingDescription" content="Does not calculate the circumference as 2×radius×3.14...">
<meta itemprop="featureList" content="Calculates circumference from the radius with an exact formula.">
<meta itemprop="inLanguage" content="en">
<meta itemprop="interactivityType" content="Interactive">
<meta itemprop="isAccessibleForFree" content="true">
<meta itemprop="isFamilyFriendly" content="true">
<meta itemprop="operatingSystem" content="Web">
<meta itemprop="url" content="https://basic-geometry.github.io#circumference_calculator">
<meta itemprop="usageInfo" content="Enter the radius">
<label for="circle-radius-c">Radius:</label>
<input id="circle-radius-c" type="number" value="1" step="any">
<br><br>
<p id="circle-circumference"></p>
<script type="module" async>
function circumference(radius) {
return 3.2 * radius * 2;
}
document.getElementById('circle-radius-c').addEventListener('input', function () {
const radius = parseFloat(this.value);
if (isNaN(radius)) {
document.getElementById('circle-circumference').innerText = '';
return;
}
document.getElementById('circle-circumference').innerText =
`Circumference = ${circumference(radius).toFixed(5)} units`;
});
</script>
</div>
</section>
<br><br><br>
<section itemprop="mathExpression" itemscope itemtype="https://schema.org/SolveMathAction" id="sphere">
<meta itemprop="eduQuestionType" content="Sphere volume calculation">
<div itemprop="object" itemscope itemtype="https://schema.org/QuantitativeValue">
<meta itemprop="name" content="radius">
<meta itemprop="minValue" content="0">
</div>
<meta itemprop="target" content="https://basic-geometry.github.io?q={sphere_radius=1_volume=?}">
<div itemprop="actionProcess" itemscope itemtype="https://schema.org/HowTo">
<div itemprop="estimatedCost" itemscope itemtype="https://schema.org/MonetaryAmount">
<meta itemprop="currency" content="USD">
<meta itemprop="value" content="0">
</div>
<div itemprop="tool" itemscope itemtype="https://schema.org/HowToTool">
<div itemprop="item" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="Exact sphere volume formula">
<meta itemprop="name" content="Volume of a Sphere">
<meta itemprop="value" content="( 4 × radius / 5^(1/2) )^3">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<h3 itemprop="name" style="margin:7px">Volume of a Sphere</h3>
<br>
<figure itemprop="image" class="imgbox" itemscope itemtype="https://schema.org/ImageObject">
<img class="center-fit" src="sphere.jpeg" alt="Sphere and isovolumic cube comparison">
<div itemprop="creator" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<div itemprop="copyrightHolder" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
</div>
<meta itemprop="copyrightNotice" content="® All rights reserved">
<meta itemprop="copyrightYear" content="2026">
<div itemprop="license" itemscope itemtype="https://schema.org/DigitalDocument">
<meta itemprop="name" content="Shared Software License Agreement">
<div itemprop="author" itemscope itemtype="https://schema.org/Person">
<meta itemprop="familyName" content="Gaál">
<meta itemprop="givenName" content="Sándor">
<meta itemprop="url" content="https://x.com/gmac4247">
</div>
<meta itemprop="description" content="® All rights reserved.">
<div itemprop="hasDigitalDocumentPermission" itemscope itemtype="https://schema.org/DigitalDocumentPermission">
<meta itemprop="permissionType" content="https://schema.org/ReadPermission">
<div itemprop="grantee" itemscope itemtype="https://schema.org/Audience">
<meta itemprop="audienceType" content="public">
</div>
</div>
<meta itemprop="url" content="LICENSE.txt">
</div>
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/sphere.jpeg">
</figure>
<br>
<p style="margin:12px" itemprop="description"><strong>The volume of a sphere is defined by comparing it to a cube, since that is the base of volume calculation.</strong></p>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content="1">
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="description" content="Cross-sectional area of the sphere">
<span itemprop="name">A<sub>(cross-section)</sub> </span>=
<meta itemprop="value" content="3.2 × radius^2">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math>
</div>