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正多边形棱柱与圆柱体积探索

正多边形棱柱与圆柱体积探索
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2020-07-04 00:10
多边形立体几何极限几何作图
①本范例演示正多边形棱柱体积的构成，通过堆叠底面图形直观展示体积计算原理；②涉及核心数学概念包括体积公式、多边形性质、极限思想及空间几何；③作图要点运用了Sequence（数列）、Polygon（多边形）、Vector（向量）、Center（中心点）、Prism（棱柱）等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者用于立体几何与极限概念的教学与探索。

GeoGebra 命令及参数

  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