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Volume of Regular Polygonal Prism and Cylinder

Volume of Regular Polygonal Prism and Cylinder
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2020-07-04 00:10
PolygonSolid GeometryLimitGeometric Construction
This example demonstrates the volume of a regular polygonal prism by stacking base pieces, showing how increasing the number of sides tends towards a cylinder. Key math concepts include volume formulas, polygon properties, limit concepts, and spatial geometry. GeoGebra commands used are Sequence, Polygon, Vector, Center, and Prism to construct the model. Suitable for high school students and math enthusiasts exploring solid geometry and limits.

GeoGebra Commands & Parameters

  1. Sequence(((r * cos(((2 * π) * i / n))), (r * sin(((2 * π) * i / n)))), i, 0, n, 1) → list1
  2. Polygon(Element[list1, 1], Element[list1, 2], n) → poly
  3. Vector((0, 0, 1)) → vUp
  4. Center(c) → A
  5. Sequence(CircleArc[A, Element[list1, i], Element[list1, i + 1]], i, 1, n, 1) → list2
  6. Sequence(Dilate[A, If[0 < a ≤ 1, 1 - a, 0], Element[list1, i]], i, 1, n, 1) → list3
  7. Sequence(Segment[A, Element[list3, i]], i, 1, n, 1) → list4
  8. Prism(poly, height) → prismOG, B, K, face40, face41, face42, face43, face44, face45, face46, face47, face48, face49, L, face50, face51, face52, face53, face54, face55, face56, face57, face58, face59, M, face60, faceBCD, edge1, edge2, edge3, edge4, edge5, edge6, edge10, edge11, N, edge12, edge13, edge14, edge15, edge22, edge23, edge24, edge25, edge26, edge32, O, edge33, edge36, edge38, edge39, edge40, edge44, edge45, edge46, edge47, edge48, P, edge49, edge50, edge51, edge52, edge53, edge64, edge65, edge66, edge67, edge68, Q, edge69, edge76, edge77, edge78, edge79, edge80, edge81, edge88, edge90, edge91, R, edge92, edge96, edge97, edge98, edge99, edge100, edge101, edge102, edge103, edge104, S, edge105, edge106, edgeBC, edgeCD, edgeBD, edge7, edge8, edge9, edge16, edge17, T, edge18, edge19, edge20, edge21, edge27, edge28, edge29, edge30, edge31, edge34, C, U, edge35, edge37, edge41, edge42, edge43, edge54, edge55, edge56, edge57, edge58, V, edge59, edge60, edge61, edge62, edge63, edge70, edge71, edge72, edge73, edge74, W, edge75, edge82, edge83, edge84, edge85, edge86, edge87, edge89, edge93, edge94, Z, edge95, edge107, edge108, edge109, edge110, edge111, edge112, edge113, edge114, edge115, A_1, edge116, edge117, B_1, C_1, D_1, E_1, F_1, D, G_1, H_1, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, E, Q_1, R_1, S_1, T_1, U_1, V_1, W_1, Z_1, A_2, B_2, F, C_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2, L_2, G, M_2, face1, face3, face2, face4, face5, face6, face7, face8, face9, H, face10, face11, face12, face13, face14, face15, face16, face17, face18, face19, I, face20, face21, face22, face23, face24, face25, face26, face27, face28, face29, J, face30, face31, face32, face33, face34, face35, face36, face37, face38, face39
  9. Sequence(Translate[poly, Vector[(If[stack / 10 < (i * ((2 * height) + maxHeight)), NaN, Max[((i + 1) * ((2 * height) + maxHeight)) - stack / 10, 0] + i] * vUp)]], i, 0, height - 1, 1) → polyPiecesMove
  10. Sequence(Prism[Element[polyPiecesMove, i], 1], i, 1, Length[polyPiecesMove], 1) → prismPieces