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‎index.html‎

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},
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"dateCreated" :"2019-01-11",
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"datePublished":"2020-01-11",
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"dateModified" :"2025-05-27",
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"dateModified" :"2025-05-28",
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"description" : "Introducing the best-established and most accurate framework to calculate area and volume.",
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"disambiguatingDescription": "Exact, empirically grounded and rigorously proven formulas over the conventional approximations.",
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"headline":"Exact geometry",
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{
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"@type": "SolveMathAction",
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"name": "Circumference of a circle",
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"description": "The exact circumference of a circle using a refined approach that replaces traditional π-based approximations. This method is based on direct shape relationships, ensuring greater accuracy in real-world measurements.",
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"description": "Algebraic derivation of the exact circumference of a circle from its area",
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"disambiguatingDescription": "Replaces traditional π-based approximations ensuring greater accuracy in real-world measurements.",
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"keywords":"Circumference of a Circle, Algebraic Derivation from the Area, Rigorously Proven",
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"target": "https://basic-geometry.github.io/",
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<div>
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<h1 style="font-size:160%;margin:7px">Exact geometry</h1>
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<br>
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<div class="imgbox">
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<figure>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<img class="center-fit" src="geometry.jpeg" alt="Geometry">
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<figcaption>
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<p>&copy; 2025 Gaál Sándor. All rights reserved.</p>
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<p>Created by: <a href="https://www.x.com/gmac4247">Gaál Sándor</a>, HU-2000 Szentendre, Ady Endre way 6.a</p>
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<p class="description">Artistic figure of the Core Geometric System ™</p>
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</figcaption>
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</figure>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="An artistic geometric illustration.">
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</div>
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<br>
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<p style="margin:12px;" >Providing the best-established and most accurate framework to calculate area and volume using the 3D coordinate system.
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<div>
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<h2 style="font-size:160%;margin:7px;">Area of a square</h2>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="square.png" alt="Square" id="square">
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<figcaption>
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<p>&copy; 2025 Gaál Sándor. All rights reserved.</p>
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<p>Created by: <a href="https://www.x.com/gmac4247">Gaál Sándor</a>, HU-2000 Szentendre, Ady Endre way 6.a</p>
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<p class="description">Area of a square</p>
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</figcaption>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Area of a square">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML" >
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<div>
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<h3 style="font-size:160%;margin:7px;">Volume of a cube</h3>
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<br>
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<div class="imgbox">
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<figure>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="cubeMarkup.jpeg" alt="Cube" id="cube">
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<figcaption>
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<p>&copy; 2025 Gaál Sándor. All rights reserved.</p>
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<p>Created by: <a href="https://www.x.com/gmac4247">Gaál Sándor</a>, HU-2000 Szentendre, Ady Endre way 6.a</p>
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<p class="description">Volume of a cube</p>
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</figcaption>
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</figure>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Volume of a cube">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
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<div>
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<h4 style="font-size:160%;margin:7px">Trigonometry</h4>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="trigonometry.png" alt="Trigonometry" id="trigonometry">
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</div>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Trigonometric entities">
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</figure>
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</div>
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<br>
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<div>
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<p style="margin:12px;">
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<div>
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<h5 style="font-size:160%;margin:7px;">Area of a regular polygon</h5>
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<br>
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<div class="imgbox">
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<img class="center-fit" src="pentagon.png" alt="Pentagon" id="pentagon">
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</div>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="pentagon.png" alt="Pentagon" id="pentagon">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Area of a regular polygon">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<div>
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<h6 style="font-size:160%;margin:7px">Area of a circle</h6>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="areaOfACircle.jpg" alt="Circle" id="circle">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact area of a circle based on direct comparison with a square.">
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</figure>
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</div>
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<br>
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<p style="margin:12px;" >The area of a circle is defined by comparing it to a square since that is the base of area calculation.
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<div>
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<h7 style="font-size:160%;margin:7px">Area of a circle segment</h7>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="circleSegment.jpg" alt="Circle-segment" id="segment">
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</div>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact area of a circle segment by subtracting a triangle from a circle slice">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The area of a circle segment can be
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calculated by subtracting a triangle from a
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<div>
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<h8 style="font-size:160%;margin:7px">Circumference of a circle</h8>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="circumference.jpg" alt="Circle" id="circumference">
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</div>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Algebraic derivation of the exact circumference of a circle from its area">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The circumference of a circle can be derived
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from its area algebraically.
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<div>
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<h9 style="font-size:160%;margin:7px;">Volume of a sphere</h9>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="sphereAndCubeMarkup.jpeg" alt="Sphere" id="sphere">
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</div>
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<br>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="Area of a square">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The volume of a sphere is defined by comparing it to a cube, as that's the base of volume calculation.
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<br><br>
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Just as the volume of a cube equals the square root of its cross sectional area cubed - <math xmlns="http://www.w3.org/1998/Math/MathML" >
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<h10 style="font-size:160%;margin:7px;">Volume of a spherical cap
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</h10>
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<br>
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<div class="imgbox">
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<img class="center-fit" src="sphericalCap.jpg" alt="Sphere" id="cap">
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</div>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="sphericalCap.jpg" alt="Sphere" id="cap">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The volume of a spherical cap based on the radius of the sphere and the cap">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">Volume of a spherical cap:
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<br>
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<div>
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<h11 style="font-size:160%;margin:7px;">Volume of a cone</h11>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="coneAndSphereMarkup.jpeg" alt="Cone-and-sphere" id="cone">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="comparing the volume of a quarter cone to an octant sphere with an equal radius">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The volume of a cone can be calculated by algebraically comparing the volume of a quarter cone with equal radius and height to an octant sphere with equal radius, through a quarter cylinder.
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</p>
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<br>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="octantSphereQuarterCone.jpeg" alt="Sphere-and-vertical-frustum-cone">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="comparing the volume of a quarter cone to an octant sphere with an equal radius">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML" >
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<p style="margin:12px;">The base of the two shapes is a quarter circle.
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</p>
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<br>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="coneAndSphereComparison.png" alt="Sphere-and-cone-projection">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="comparing the volume of a quarter cone to an octant sphere with an equal radius">
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</figure>
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</div>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="sphereAndConeMarkup.jpeg" alt="Sphere-and-cone">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="comparing the volume of a quarter cone to an octant sphere with an equal radius">
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</figure>
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</div>
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<br>
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<p style="margin:12px;"><math xmlns="http://www.w3.org/1998/Math/MathML" >
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<div>
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<h12 style="font-size:160%;margin:7px;">Volume of a frustum cone</h12>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="frustumOfConeMarkup.png" alt="Horizontal-frustum-cone">
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</div>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact volume of a frustum cone based on its top and bottom diameter and height">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The volume of a frustum cone can be
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calculated by subtracting the missing tip
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<div>
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<h13 style="font-size:160%;margin:7px;">Surface area of a cone</h13>
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<br>
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<div class="imgbox">
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<img class="center-fit" src="coneMarkup.jpeg" alt="Cone" id="coneSurface">
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</div>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="coneMarkup.jpeg" alt="Cone" id="coneSurface">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact surface area of a cone based on its radius and height">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML" >
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<mrow>
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<div>
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<h14 style="font-size:160%;margin:7px">Volume of a pyramid</h14>
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<br>
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<div class="imgbox">
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<img class="center-fit" src="conePyramidVolumeMarkup.jpeg" alt="Pyramids" id="pyramid">
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</div>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="conePyramidVolumeMarkup.jpeg" alt="Pyramids" id="pyramid">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact volume of a pyramid based on its bottom area and height using the coefficient of the volume of a cone">
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</figure>
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</div>
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<br>
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<br>
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<div class="imgbox">
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<div>
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<h15 style="font-size:160%;margin:7px;">Volume of a horizontal frustum pyramid</h15>
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<br>
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<div class="imgbox">
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="frustumOfPyramidMarkup.png" alt="Horizontal-frustum-pyramid">
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</div>
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact volume of a frustum pyramid based on its top and bottom base area and height">
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</figure>
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</div>
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<br>
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<p style="margin:12px;">The volume of a frustum pyramid can be
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calculated by subtracting the missing tip
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<div>
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<h16 style="font-size:160%;margin:7px">Volume of a tetrahedron</h16>
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<br>
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<div class="imgbox">
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<img class="center-fit" src="tetrahedronMarkup.jpeg" alt="Tetrahedron" id="tetrahedron">
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</div>
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<div class="imgbox" itemscope itemtype="http://schema.org/CreativeWork">
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<figure>
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<img class="center-fit" src="tetrahedronMarkup.jpeg" alt="Tetrahedron" id="tetrahedron">
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<meta itemprop="creator" content="Gaál Sándor HU-2000 Szentendre Ady Endre way 6.A">
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<meta itemprop="copyrightYear" content="2025">
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<meta itemprop="description" content="The exact volume of a tetrahedron based on its edge length">
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</figure>
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</div>
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<br>
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<math style="margin:12px;" xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>

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