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<title>Home of Basic Geometry</title>
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<main id="cgs">
<h1 style="font-size:130%;margin:7px">Featuring The Core Geometric System ™</h1>
<br><br>
<section id="circle">
<h2 style="margin:7px">Area of a Circle</h2>
<br>
<figure class="imgbox">
<img class="center-fit" src="circle.png" alt="Circle area">
</figure>
<br>
<p style="margin:12px"><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">
<p>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">
<p>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>
<span>radius² </span>=
<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>
<br>
<div style="margin:12px">
<span>radius </span>=
<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 style="margin:12px">
<p>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>
<span>A<sub>(circle)</sub> </span>=
<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">
<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 id="circumference">
<h3 style="margin:7px">Circumference of a Circle</h3>
<br>
<figure class="imgbox">
<img class="center-fit" src="circumference.png" alt="The circumference of a circle is derived from its area algebraically by subtracting a smaller circle">
</figure>
<br>
<div style="margin:12px">
<p><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>
<br><br>
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>
<span >Circumference </span>=
<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 >
<div>
<p>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>
<p>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>
<br><br><br>
<div>
<p>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>
<br><br><br>
<div>
<p>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>
<br><br><br>
<div>
<p>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>
<br><br><br>
<div>
<p>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>
<br><br><br>
<div>
<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>
<br>
<div style="margin:12px">
<p>The length of the circumference approaches 6.4 × radius as its thickness approaches 0.</p>
<br><br>
<span>Circumference </span>=
<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">
<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 id="sphere">
<h3 style="margin:7px">Volume of a Sphere</h3>
<br>
<figure class="imgbox">
<img class="center-fit" src="sphere.jpeg" alt="Sphere and isovolumetric cube comparison">
</figure>
<br>
<p style="margin:12px"><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">
<span>A<sub>(cross-section)</sub> </span>=
<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>
<br><br>
<div style="margin:12px">
<p>Take the square root of the cross-sectional area.</p>
<br>
<span>√A<sub>(cross-section)</sub> </span>=
<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>
<br>
<div style="margin:12px">
<p>The volume of a sphere equals the cubic value of the square root of its cross-sectional area, just like a cube.</p>
<br><br>
<span>V<sub>(sphere)</sub> </span>=
<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>
<br><br>
<div style="margin:12px" id="sphere_volume_calculator">
<label for="sphere-radius">Radius:</label>
<input id="sphere-radius" type="number" value="1" step="any">
<br><br>
<p id="sphere-volume"></p>
<script type="module" async>
function sphereVolume(radius) {
return Math.pow(Math.sqrt(3.2) * radius, 3);
}
document.getElementById('sphere-radius').addEventListener('input', function () {
const radius = parseFloat(this.value);
if (isNaN(radius)) {
document.getElementById('sphere-volume').innerText = '';
return;
}
document.getElementById('sphere-volume').innerText =
`Volume = ${sphereVolume(radius).toFixed(5)} cubic units`;
});
</script>
</div>
</section>
<br><br><br>
<section style="border: 2px solid red;
padding: 10px" id="sphere-surface">
<p style="margin:6px">Advertisement</p>
<br><br>
<h3 style="margin:7px">Surface Area of a Sphere</h3>
<br>
<div class="imgbox">
<img class="center-fit" src="sphereSurface.jpeg" alt="sphere surface illustration">
<br>
<p style="margin:6px">The image is an illustration.</p>
</div>
<br>
<p style="margin:6px">The conventional formula for the surface area of a sphere was allegedly developed from the conventional volume formula.</p>
<br>
<p style="margin:6px"><strong>
Unlock the true formula to calculate the surface area of a sphere.</strong></p>
<span style="margin:6px">$</span>
<span>3,200,000,000.00</span>
<p style="margin:6px">( + tax, if applicable )</p>
<br>
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</section>
<br><br><br><br>
<section id="cone">
<h3 style="margin:7px">Volume of a Cone</h3>
<br>
<figure class="imgbox">
<img class="center-fit" src="coneAndSphere.jpeg" alt="Sphere and cone">
</figure>
<br>
<p style="margin:12px"><strong>The volume of a cone can be calculated by algebraically comparing the volume of a vertical quadrant of a cone with equal radius and height to an octant sphere with equal radius, through a quadrant cylinder.</strong></p>
<br>
<figure class="imgbox">
<img class="center-fit" src="octantSphereQuarterCone.jpeg" alt="Comparison">
</figure>
<br>
<div style="margin:12px">
<span>V<sub>(octant sphere)</sub> </span>=
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mrow>
<mfrac>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<msqrt>
<mn>3.2</mn>
</msqrt>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3.2</mn>
</msqrt>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3.2</mn>
</msqrt>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>)</mo>
<mo>×</mo>
<mo>(</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3.2</mn>
</msqrt>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>)</mo>
<mo>×</mo>
<mo>(</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3.2</mn>
</msqrt>
<mo>⁢</mo> <!-- invisible times -->
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>)</mo>
</mrow>
</mrow>
</math>
</div>
<br>
<figure class="imgbox">
<img class="center-fit" src="sphereAndCone.jpeg" alt="Sphere and cone">
</figure>
<br>
<div style="margin:12px">
<p>The base of the two shapes is a quadrant circle.</p>
<br>
<math xmlns="http://www.w3.org/1998/Math/MathML" >
<mrow>
<msub>
<mi>A</mi>
<mi>(base)</mi>