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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">
<link rel="canonical" href="https://basic-geometry.pages.dev">
<link rel="icon" type="image/png" sizes="256x256" href="android-chrome-256x256.png">
<link rel="icon" type="image/png" sizes="192x192" href="android-chrome-192x192.png">
<link rel="icon" type="image/png" sizes="180x180" href="apple-touch-icon.png">
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<link rel="icon" type="image/png" sizes="16x16" href="favicon-16x16.png">
<link rel="icon" type="image/x-icon" href="favicon.ico">
<link rel="mask-icon" href="safari-pinned-tab.svg" color="#000000">
<link rel="manifest" href="manifest.json">
<link href="https://fonts.googleapis.com/css2?family=OpenDyslexic&display=swap" rel="stylesheet">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<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">
<meta name="twitter:card" content="summary_large_image">
<meta name="twitter:creator" content="@gmac4247">
<meta name="twitter:site" content="https://basic-geometry.github.io">
<meta name="twitter:title" content="Home of Basic Geometry">
<meta name="twitter:description" content="Universally applicable self-consistent geometric framework of physically grounded exact area and volume calculation formulas for regular shapes.">
<meta name="twitter:image" content="android-chrome-256x256.png">
<meta name="msapplication-TileColor" content="#000000">
<meta name="theme-color" content="#000000">
<style>
* {
margin: 0;
padding: 0;
}
.imgbox {
display: grid;
height: 100%;
}
.center-fit {
max-width: 100%;
max-height: 100vh;
margin: auto;
}
#cookie-notice {
position: fixed;
top: 0;
left: 0;
right: 0;
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;
}
#cookie-notice-text {
margin-right: 20px;
}
#cookie-notice-buttons {
display: flex;
align-items: center;
}
#cookie-notice-buttons button {
margin-left: 10px;
background-color: #4CAF50;
color: white;
border: none;
padding: 10px;
border-radius: 5px;
cursor: pointer;
}
#cookie-notice-buttons button:hover {
background-color: #3e8e41;
}
.rounded-button {
display: inline-block;
padding: 7px 9px;
text-decoration: none;
border-radius: 9px;
text-align: center;
transition: background-color 0.3s ease;
}
body {
font-family: 'OpenDyslexic', sans-serif;
}
</style>
</head>
<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.
Love it, or leave it.
</div>
<div id="cookie-notice-buttons">
<button onclick="document.getElementById('cookie-notice').style.display = 'none';">Accept</button>
<button onclick="location.replace('about:blank')">Leave</button>
</div>
</div>
<main itemscope itemtype="https://schema.org/LearningResource https://schema.org/MathSolver" id="cgs">
<meta itemprop="name" content="Core Geometric System ™">
<meta itemprop="dateModified" content="2026-08-26T21:15:29+02:00">
<meta itemprop="accessMode" content="mathOnVisual">
<meta itemprop="accessMode" content="textual">
<meta itemprop="accessMode" content="visual">
<meta itemprop="accessibilityControl" content="fullKeyboardControl">
<meta itemprop="accessibilityControl" content="fullTouchControl">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Read text and equations with figures, calculate.">
<div itemprop="audience" itemscope itemtype="https://schema.org/EducationalAudience">
<meta itemprop="educationalRole" content="For teachers and learners">
</div>
<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">
<link itemprop="license" href="https://basic-geometry.pages.dev">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<div itemprop="educationalAlignment" itemscope itemtype="https://schema.org/AlignmentObject">
<meta itemprop="alignmentType" content="educationalSubject">
<meta itemprop="educationalFramework" content="Geometric framework">
<meta itemprop="targetName" content="Core Geometric System ™">
<meta itemprop="targetUrl" content="https://basic-geometry.github.io#cgs">
</div>
<meta itemprop="educationalLevel" content="from basic to advanced">
<meta itemprop="inLanguage" content="en">
<meta itemprop="isAccessibleForFree" content=true>
<meta itemprop="isFamilyFriendly" content=true>
<meta itemprop="keywords" content="Core Geometric System, Exact Geometric Calculations, Analysis, Engineering Design Solutions, Computer Graphics Rendering, Algorithm Optimization, Navigation">
<meta itemprop="mainEntityOfPage" content="https://basic-geometry.github.io">
<meta itemprop="schemaVersion" content="https://schema.org/docs/releases.html#v30.0">
<meta itemprop="timeRequired" content="PT8H">
<meta itemprop="typicalAgeRange" content="5-105">
<meta itemprop="url" content="https://basic-geometry.github.io">
<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">
<h2 itemprop="name" style="margin:7px">Area of a Circle</h2>
<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">
<meta itemprop="name" content="derive the Area of a Circle">
<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">
<meta itemprop="name" content="The Right Triangle Hypotenuse Theorem formula">
<div itemprop="item" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="name" content="hypotenuse^2">
<meta itemprop="value" content="(leg1)^2 + (leg2)^2">
<meta itemprop="description" content="The Right Triangle Hypotenuse Theorem">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<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>
<meta itemprop="copyrightNotice" content="® All rights reserved">
<meta itemprop="copyrightYear" content="2026">
<meta itemprop="creditText" content="Made with the help of Vectorink.io">
<link itemprop="license" href="https://basic-geometry.pages.dev">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/circle.png">
</figure>
<br>
<span style="margin:12px" itemprop="abstract"><strong>The area of a circle is defined by comparing it to a square since that is the base of area calculation.</strong></span>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=1>
<span itemprop="abstract">The circle can be cut into four quadrants, each placed with their origin on the vertices of a square.</span>
<br><br>
<ul itemprop="abstract">
<li>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.</li>
<li>The arcs of the quadrants of a circumscribed circle would overlap, and intersect at the center of the square, covering it all.</li>
</ul>
<br>
<span itemprop="abstract"><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></span>
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=2>
<span 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.</span>
<br><br>
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">radius² </span>=
<meta itemprop="value" content="( side / 4 )^2 + ( side / 2 )^2">
<meta itemprop="description" content="The ratio between the squared radius of the circle and an equiareal square">
<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">
<span itemprop="name">radius </span>=
<meta itemprop="value" content="side × 5^(1/2) / 4">
<meta itemprop="description" content="The ratio between the radius of the circle and an equiareal square">
<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/DefinedTerm">
<meta itemprop="name" content="Area of a circle">
<span 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.</span>
<br><br><br>
<div itemprop="identifier" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">A<sub>circle</sub> </span>=
<meta itemprop="value" content="3.2 × radius^2">
<meta itemprop="description" content="Exact circle area formula">
<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>
</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="accessibilityControl" content="fullTouchControl">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Enter numbers and read result">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<div itemprop="aggregateRating" itemscope itemtype="https://schema.org/AggregateRating">
<meta itemprop="bestRating" content=9>
<meta itemprop="worstRating" content=0>
<meta itemprop="ratingValue" content=9>
<meta itemprop="reviewCount" content=1>
</div>
<meta itemprop="applicationCategory" content="Geometric Calculator">
<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>
<link itemprop="license" href="https://basic-geometry.pages.dev">
<div itemprop="offers" itemscope itemtype="https://schema.org/Offer">
<meta itemprop="name" content="Personal, non-commercial use">
<meta itemprop="description" content="Non-exclusive, non-transferable">
<meta itemprop="price" content=0>
<meta itemprop="priceCurrency" content="USD">
</div>
<meta itemprop="operatingSystem" content="Web">
<meta itemprop="url" content="https://basic-geometry.github.io#circle_area_calculator">
<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">
<h3 itemprop="name" style="margin:7px">Circumference of a Circle</h3>
<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">
<meta itemprop="name" content="derive the Circumference of a Circle">
<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">
<meta itemprop="name" content="Circle area formula">
<meta itemprop="description" content="Exact, physical area of a circle.">
<meta itemprop="description" content="Direct circle-to-square comparison.">
<meta itemprop="disambiguatingDescription" content="Not the traditional, abstract pi≈3.14... approximate.">
<div itemprop="identifier" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="name" content="Area of a circle">
<meta itemprop="value" content="3.2 × radius^2">
<meta itemprop="description" content="Universally applicable formula to calculate the exact physical area of a circle in terms of its radius.">
<meta itemprop="description" content="Derived via cutting the circle into four quadrants, placing them with their origin on the vertices of a square with side = 4radius / 5^(1/2).">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<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">
<meta itemprop="creditText" content="Made with the help of Vectorink.io">
<link itemprop="license" href="https://basic-geometry.pages.dev">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/circumference.png">
</figure>
<br>
<span style="margin:12px" itemprop="abstract"><strong>The circumference of a circle is derived algebraically from its area by subtracting a circle with a radius smaller than the radius of the actual circle by the virtual thickness of the circumference — which distinguishes it from the rest of the circle —, and dividing the difference by that thickness.</strong></span>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=1>
<span itemprop="abstract">Take a circle with radius = r. (A = 3.2r²).</span>
<meta itemprop="description" content="Instantiate a circle object">
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=2>
<span itemprop="abstract">Let x represent the virtual thickness of the circumference.
<br>
Subtract an inner circle with radius = r - x, (A = 3.2(r - x)²) to generate the boundary region entity.</span>
<meta itemprop="description" content="Define the circumference object as a ring, or annulus.">
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=3>
<span itemprop="abstract">As x approaches 0, the difference between the areas of the ring representing the circumference and a rectangle with sides equal to the circumference and its thickness becomes negligible.</span>
<meta itemprop="description" content="Normalize Dimension">
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=4>
<span itemprop="abstract">The length of the two shorter sides of the rectangle is x.</span>
<meta itemprop="description" content="Map Dimension">
</div>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=5>
<span itemprop="abstract">The length of the two longer sides is the area of the rectangle divided by x.</span>
<meta itemprop="description" content="Render the circumference object into a property of the circle object: Transition the output from an instance of a 2D spatial region into a scalar property attached to the parent circle.">
</div>
<br>
<div style="margin:12px" itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">Circumference </span>=
<meta itemprop="value" content="( 3.2 × radius^2 - 3.2 × ( radius - circumferenceWidth )^2 ) / circumferenceWidth">
<meta itemprop="description" content="Circumference of a circle in terms of radius and thickness.">
<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>
<br>
<details style="margin:12px">
<summary><h4>Expand</h4></summary>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=6>
<span itemprop="description">the term (r - x)²:</span>
<br><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>
<br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=7>
<span itemprop="description">Substitute this back into the original expression:</span>
<br><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>
<br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=8>
<span itemprop="description">Distribute the 3.2 inside the parentheses:</span>
<br><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>
<br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=9>
<span itemprop="description">Simplify the numerator:</span>
<br><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>
<br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=10>
<span itemprop="description">Factor out x from the numerator:</span>
<br><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>
<br><br>
<div itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=11>
<span itemprop="description">Cancel out the x in the numerator and denominator:</span>
<br><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>
<br><br>
<div>
<span>In calculus terms:</span>
<br><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>
</details>
</div>
<br>
<div style="margin:12px" itemprop="result" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Circumference of a circle">
<span itemprop="description">The length of the circumference approaches 6.4 × radius as its thickness approaches 0.</span>
<br><br><br>
<div itemprop="identifier" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">Circumference </span>=
<meta itemprop="value" content="6.4 × radius">
<meta itemprop="description" content="Exact circumference formula">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mn>6.4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>radius</mi>
</mrow>
</math>
</div>
</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="accessibilityControl" content="fullTouchControl">
<meta itemprop="accessibilityHazard" content="none">
<meta itemprop="accessibilitySummary" content="Enter numbers and read results">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<div itemprop="aggregateRating" itemscope itemtype="https://schema.org/AggregateRating">
<meta itemprop="bestRating" content=9>
<meta itemprop="worstRating" content=0>
<meta itemprop="ratingValue" content=9>
<meta itemprop="reviewCount" content=1>
</div>
<meta itemprop="applicationCategory" content="Geometric Calculator">
<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="Active">
<meta itemprop="isAccessibleForFree" content=true>
<meta itemprop="isFamilyFriendly" content=true>
<link itemprop="license" href="https://basic-geometry.pages.dev">
<div itemprop="offers" itemscope itemtype="https://schema.org/Offer">
<meta itemprop="name" content="Personal, non-commercial use">
<meta itemprop="description" content="Non-exclusive, non-transferable">
<meta itemprop="price" content=0>
<meta itemprop="priceCurrency" content="USD">
</div>
<meta itemprop="operatingSystem" content="Web">
<meta itemprop="url" content="https://basic-geometry.github.io#circumference_calculator">
<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">
<h3 itemprop="name" style="margin:7px">Volume of a Sphere</h3>
<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">
<meta itemprop="name" content="Derive the volume of a sphere">
<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">
<meta itemprop="name" content="Circle area formula">
<meta itemprop="description" content="Exact, physical area of a circle.">
<meta itemprop="description" content="Direct circle-to-square comparison.">
<meta itemprop="disambiguatingDescription" content="Not the traditional, abstract pi≈3.14... approximate.">
<div itemprop="item" itemscope itemtype="https://schema.org/PropertyValue">
<meta itemprop="name" content="Area of a circle">
<meta itemprop="value" content="3.2 × radius^2">
<meta itemprop="description" content="Universally applicable formula to calculate the exact physical area of a circle in terms of its radius.">
<meta itemprop="description" content="Derived via cutting the circle into four quadrants, placing them with their origin on the vertices of a square with side = 4radius / 5^(1/2).">
</div>
</div>
<meta itemprop="totalTime" content="PT30M">
<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">
<meta itemprop="creditText" content="Made with the help of Microsoft Copilot">
<link itemprop="license" href="https://basic-geometry.pages.dev">
<meta itemprop="acquireLicensePage" content="https://basic-geometry.github.io/privacy-policy">
<meta itemprop="contentUrl" content="https://basic-geometry.github.io/sphere.jpeg">
</figure>
<br>
<span style="margin:12px" itemprop="abstract"><strong>The volume of a sphere is defined by comparing it to a cube, since that is the base of volume calculation.</strong></span>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=1>
<span itemprop="description">The middle cross-section of a sphere is a circle.</span>
<br><br>
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">A<sub>cross-section</sub> </span>=
<meta itemprop="value" content="3.2 × radius^2">
<meta itemprop="description" content="Cross-sectional area of the sphere">
<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>
</div>
<br>
<div style="margin:12px" itemprop="step" itemscope itemtype="https://schema.org/HowToStep">
<meta itemprop="position" content=2>
<span itemprop="description">Take the square root of the cross-sectional area.</span>
<br><br>
<div itemprop="about" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">√A<sub>cross-section</sub> </span>=
<meta itemprop="value" content="(3.2)^(1/2) × radius = 4 × radius / 5^(1/2)">
<meta itemprop="description" content="Square root of the cross-sectional area of the sphere">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msqrt>
<mrow>
<mn>3.2</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow>
<msqrt>
<mrow>
<mfrac>
<mrow>
<mn>16</mn>
<mo>⁢</mo> <!-- invisible times -->
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mn>5</mn>
</mfrac>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math>
</div>
</div>
</div>
<br>
<div style="margin:12px" itemprop="result" itemscope itemtype="https://schema.org/DefinedTerm">
<meta itemprop="name" content="Volume of a sphere">
<span itemprop="description">The volume of a sphere equals the cubic value of the square root of its cross-sectional area, just like a cube.</span>
<br><br><br>
<div itemprop="identifier" itemscope itemtype="https://schema.org/PropertyValue">
<span itemprop="name">V<sub>sphere</sub> </span>=
<meta itemprop="value" content="( 4 × radius / 5^(1/2) )^3">
<meta itemprop="description" content="Exact sphere volume formula">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mfrac>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</math>
</div>
</div>
<br><br>
<div style="margin:12px" id="sphere_volume_calculator" itemprop="subjectOf" itemscope itemtype="https://schema.org/SoftwareApplication">
<meta itemprop="name" content="Sphere volume calculator">