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Golden Prism: V=TSA Solution

Golden Prism: V=TSA Solution
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2016-10-06 03:22
PolygonSolid GeometryOptimizationGeometric Construction
In collections Geo&Alg:Area Volume Pythagorean Th.
① This example demonstrates a special golden prism where the volume (V) equals the total surface area (TSA), and provides the geometric construction and numerical solution satisfying this condition. ② Core mathematical concepts include volume and surface area of prisms, the golden ratio (geometric mean), and solving equations relating geometric quantities. ③ Construction highlights: using GeoGebra commands such as Segment, Polygon, Midpoint, Ray, Circle, Intersect, OrthogonalLine, Prism, Volume, Point, the example builds polygonal bases, lateral faces, vertices and edges, and determines key dimensions by equating volume and surface area. ④ Suitable for high school or early college solid geometry lessons, exploring special properties of the golden ratio in solids, or as a mathematical modeling and equation-solving exercise.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Polygon(A, B, 4) → poly1, g, h, i, j, C, D
  3. Midpoint(h) → E
  4. Segment(E, D) → k
  5. Ray(B, C) → l
  6. Circle(E, k) → c
  7. Intersect(c, l) → F,
  8. OrthogonalLine(F, l) → m
  9. Ray(A, D) → n
  10. Intersect(m, n) → G
  11. Polygon(A, G, F, B) → poly2, a, g_1, f_1, w
  12. Prism(poly2, a) → d, H, edgeAH, edgeGI, edgeFJ, edgeBK, edgeHI, edgeIJ, edgeJK, edgeHK, I, J, K, faceAGIH, faceFGIJ, faceBFJK, faceABKH, faceHIJK
  13. Volume(d) → volumed
  14. Point(d) → Pointd
  15. Point(Circle[B, 20^(0.5)]) → L
  16. Segment(B, L) → sqrt20