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193 | 193 | }, |
194 | 194 | "dateCreated": "2024-08-31", |
195 | 195 | "datePublished": "2024-08-31", |
196 | | -"dateModified": "2025-11-11", |
| 196 | +"dateModified": "2025-11-14", |
197 | 197 | "description": "About the context of the Core Geometric System ™, the best-established and most accurate framework to calculate area and volume.", |
198 | 198 | "disambiguatingDescription": "Exact, empirically grounded and rigorously proven formulas over the conventional approximations.", |
199 | 199 | "headline": "Introducing the Core Geometric System ™", |
@@ -1270,7 +1270,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2> |
1270 | 1270 | <p style="margin:12px;">The commonly used base × height / 3 approximation for the volume of a pyramid was likely estimated based on two observations. |
1271 | 1271 | <br> |
1272 | 1272 | <br> |
1273 | | -One is that the area of the mid-height cross section of a regular pyramid - of which's apex can be connected to the midpoint of the base with a perpendicular line - is exactly a quarter of a circumscribed solid's with the same base and height. |
| 1273 | +One is that the area of the mid-height cross section of a regular pyramid — of which's apex can be connected to the midpoint of the base with a perpendicular line — is exactly a quarter of a circumscribed solid's with the same base and height. |
1274 | 1274 | <br> |
1275 | 1275 | That makes the ratio between the mid-height cross-sectional area of the pyramid, and the difference between the mid-height cross-sectional areas of the circumscribed solid and the pyramid 1 : 3 . |
1276 | 1276 | </p> |
@@ -1365,7 +1365,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2> |
1365 | 1365 | The same is true for a cone. |
1366 | 1366 | <br> |
1367 | 1367 | <br> |
1368 | | -Can this ratio can be generalized for the overall volume of any cone and pyramid? |
| 1368 | +Can this ratio be generalized for the overall volume of any cone and pyramid? |
1369 | 1369 | <br> |
1370 | 1370 | <br> |
1371 | 1371 | No. Because it's not true in case of most other shapes. |
@@ -1459,7 +1459,7 @@ <h2 style="margin:6px;">SURFACE AREA OF A SPHERE</h2> |
1459 | 1459 | - height is e, |
1460 | 1460 | <br> |
1461 | 1461 | <br> |
1462 | | -then the volume of each pyramid has to be larger than 1/3 × base × height, because 3 such pyramids can't form a cube with the same edge length, because their vertices and faces can't occupy the same space simultaneously. |
| 1462 | +then the volume of each pyramid has to be larger than 1 / 3 × base × height, because 3 such pyramids can't form a cube with the same edge length, because their vertices and faces can't occupy the same space simultaneously. |
1463 | 1463 | <br> |
1464 | 1464 | <br> |
1465 | 1465 | The vertices are the most obvious examples, but the same is true for the edges, the diagonals and the inner faces. |
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