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三角形、平行四边形、梯形和菱形的面积

三角形、平行四边形、梯形和菱形的面积
几何画板及绘图内容加载中，请稍等……
2015-07-27 07:38
多边形变换几何作图滑块动画
本范例演示如何计算特定多边形的面积。通过移动滑块，可以看到这些形状如何“转变”为其他形状。涉及的核心数学概念包括相似变换、面积计算等。使用了Polygon, Midpoint, Segment, OrthogonalLine, Intersect, Rotate, Vector, Line, Point, Angle, Translate, Mirror等GeoGebra命令。适合初中及以上学生学习几何变换和面积计算。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Midpoint(B, A) → D
  3. Midpoint(A, C) → E
  4. Segment(D, E) → d
  5. OrthogonalLine(A, d) → e
  6. Intersect(e, d) → F
  7. Polygon(A, F, D) → poly2, d_1, a_1, f
  8. Polygon(F, E, A) → poly3, a_2, f_1, e_1
  9. Rotate(A, Move, D) → A'
  10. Rotate(F, Move, D) → F'
  11. Rotate(D, Move, D) → D'
  12. Polygon(A', F', D') → poly2', d', a', f'
  13. Rotate(F, (-Move), E) → F'_1
  14. Rotate(E, (-Move), E) → E'
  15. Rotate(A, (-Move), E) → A'_1
  16. Polygon(F'_1, E', A'_1) → poly3', a'_1, f'_1, e'
  17. OrthogonalLine(A, a) → g
  18. Intersect(a, g) → G
  19. Midpoint(A, G) → H
  20. Vector(H, A) → u
  21. Vector(H, G) → v
  22. Line(A, a) → h
  23. OrthogonalLine(B, h) → i
  24. OrthogonalLine(C, h) → j
  25. Intersect(h, i) → I
  26. Intersect(h, j) → J
  27. Polygon(I, J, C, B) → poly5, i_1, j_1, c_2, b_2
  28. Point(i) → K
  29. OrthogonalLine(K, j) → k
  30. Intersect(j, k) → L
  31. Midpoint(K, L) → M
  32. Vector(M, K) → w
  33. Vector(M, L) → l
  34. Angle(B, A, C) → α
  35. Angle(A, C, B) → β
  36. Angle(C, B, A) → γ
  37. Line(C, B) → p
  38. OrthogonalLine(A, p) → q
  39. Line(C, q) → r
  40. Line(A, p) → t
  41. Intersect(p, q) → N
  42. Intersect(r, t) → O
  43. Line(B, r) → g_1
  44. Midpoint(A, C) → P
  45. Intersect(g_1, b) → Q
  46. Line(P, a) → h_1
  47. Intersect(c, h_1) → R
  48. Segment(R, P) → k_1
  49. Segment(Q, B) → l_1
  50. Intersect(k_1, l_1) → S
  51. Segment(S, B) → m_1
  52. Polygon(A, R, P) → poly7, p_1, a_4, r_1
  53. Polygon(R, B, S) → poly8, s_1, r_2, b_3
  54. Intersect(m_1, s_1) → T
  55. Polygon(S, T, C, P) → poly9, s_2, t_1, c_4, p_2
  56. Rotate(A, (-Move), P) → A'_2
  57. Rotate(R, (-Move), P) → R'
  58. Rotate(P, (-Move), P) → P'
  59. Polygon(A'_2, R', P') → poly7', p', a'_2, r'
  60. Vector(T, C) → u_1
  61. Translate(R, Vector[(Move / π * u_1)]) → R'_1
  62. Translate(B, Vector[(Move / π * u_1)]) → B'
  63. Translate(S, Vector[(Move / π * u_1)]) → S'
  64. Polygon(R'_1, B', S') → poly8', s', r'_1, b'
  65. OrthogonalLine(T, t) → n_1
  66. Intersect(t, n_1) → U
  67. Polygon(U, O, C, T) → poly6, u_2, o_1, c_3, t_2
  68. OrthogonalLine(B, q) → q_1
  69. Intersect(q, q_1) → V
  70. Midpoint(A, V) → W
  71. Vector(W, A) → v_1
  72. Vector(W, V) → w_1
  73. Midpoint(B, A) → Z
  74. Segment(E, Z) → n
  75. OrthogonalLine(C, n) → f_2
  76. Intersect(f_2, n) → A_1
  77. Segment(A_1, C) → g_2
  78. Polygon(B, Z, A_1, C) → poly10, b_4, z_1, a_3, c_5
  79. Polygon(Z, E, A) → poly11, a_5, z_2, e_3
  80. Polygon(E, C, A_1) → poly12, a_6, e_4, c_6
  81. Rotate(Z, Move, Z) → Z'
  82. Rotate(E, Move, Z) → E'_1
  83. Rotate(A, Move, Z) → A'_3
  84. Polygon(Z', E'_1, A'_3) → poly11', a'_3, z', e'_1
  85. Translate(E, Vector[(Move / π * (w - l))]) → E'_2
  86. Translate(C, Vector[(Move / π * (w - l))]) → C'
  87. Translate(A_1, Vector[(Move / π * (w - l))]) → B_1
  88. Polygon(E'_2, C', B_1) → poly12', b_5, e'_2, c'
  89. OrthogonalLine(B, n) → b_1
  90. Intersect(b_1, n) → C_1
  91. Polygon(B, C_1, E, C) → poly4, c_1, d_2, e_2, h_2
  92. Polygon(Z, B, C, E) → poly13, i_2, j_2, k_2, l_2
  93. Segment(D_1, E_1) → m_2
  94. Segment(D_1, F_1) → n_2
  95. Line(F_1, m_2) → q_2
  96. Line(E_1, n_2) → d_3
  97. Intersect(q_2, d_3) → G_1
  98. Segment(E_1, G_1) → f_3
  99. Segment(G_1, F_1) → g_3
  100. OrthogonalLine(D_1, g_3) → h_3
  101. Intersect(h_3, g_3) → H_1
  102. Polygon(D_1, H_1, F_1) → poly14, f_4, d_4, h_4
  103. Polygon(H_1, G_1, E_1, D_1) → poly15, h_5, g_4, e_5, d_5
  104. Polygon(D_1, F_1, G_1, E_1) → poly16, d_6, f_5, g_5, e_6
  105. Vector(F_1, G_1) → v_2
  106. Translate(D_1, Vector[(Move / π * v_2)]) → I_1
  107. Translate(H_1, Vector[(Move / π * v_2)]) → J_1
  108. Translate(F_1, Vector[(Move / π * v_2)]) → K_1
  109. Polygon(I_1, J_1, K_1) → poly14', k_3, i_3, j_3
  110. Angle(G_1, F_1, D_1) → δ
  111. Angle(E_1, G_1, F_1) → ε
  112. OrthogonalLine(E_1, q_2) → l_3
  113. Intersect(q_2, l_3) → L_1
  114. Polygon(D_1, H_1, L_1, E_1) → poly17, d_7, h_6, l_4, e_7
  115. OrthogonalLine(F_1, e_5) → m_3
  116. Intersect(m_3, e_5) → M_1
  117. Polygon(M_1, F_1, D_1) → poly18, d_8, m_4, f_6
  118. Polygon(M_1, E_1, G_1, F_1) → poly19, m_5, e_8, g_6, f_7
  119. Translate(M_1, Vector[(Move / π * v_2)]) → N_1
  120. Translate(F_1, Vector[(Move / π * v_2)]) → O_1
  121. Translate(D_1, Vector[(Move / π * v_2)]) → P_1
  122. Polygon(N_1, O_1, P_1) → poly18', p_3, n_3, o_3
  123. Line(M_1, E_1) → q_3
  124. OrthogonalLine(G_1, q_3) → r_3
  125. Intersect(q_3, r_3) → Q_1
  126. Polygon(M_1, F_1, G_1, Q_1) → poly20, m_6, f_8, g_7, q_4
  127. Midpoint(D_1, H_1) → R_1
  128. Vector(R_1, D_1) → w_2
  129. Vector(R_1, H_1) → u_3
  130. Midpoint(M_1, F_1) → S_1
  131. Vector(S_1, M_1) → v_3
  132. Vector(S_1, F_1) → w_3
  133. Line(M_1, F_1) → s_3
  134. Line(Q_1, G_1) → t_3
  135. Point(s_3) → T_1
  136. OrthogonalLine(T_1, t_3) → i_4
  137. Intersect(t_3, i_4) → U_1
  138. Midpoint(T_1, U_1) → V_1
  139. Vector(V_1, T_1) → u_4
  140. Vector(V_1, U_1) → v_4
  141. Point(h_3) → W_1
  142. OrthogonalLine(W_1, l_3) → j_4
  143. Intersect(l_3, j_4) → Z_1
  144. Midpoint(W_1, Z_1) → A_2
  145. Vector(A_2, W_1) → w_4
  146. Vector(A_2, Z_1) → k_4
  147. Line(B_2, C_2) → n_4
  148. Line(D_2, n_4) → p_4
  149. Point(p_4) → E_2
  150. Segment(D_2, B_2) → r_4
  151. Segment(E_2, C_2) → s_4
  152. Segment(D_2, E_2) → t_4
  153. Segment(B_2, C_2) → i_5
  154. Point(n_4) → F_2
  155. OrthogonalLine(F_2, p_4) → j_5
  156. Intersect(p_4, j_5) → G_2
  157. Midpoint(E_2, C_2) → H_2
  158. OrthogonalLine(B_2, p_4) → k_5
  159. Point(k_5) → L_2
  160. OrthogonalLine(L_2, k_5) → l_5
  161. OrthogonalLine(C_2, l_5) → n_5
  162. Intersect(l_5, n_5) → M_2
  163. Midpoint(G_2, F_2) → K_2
  164. Midpoint(L_2, M_2) → N_2
  165. OrthogonalLine(D_2, i_5) → p_5
  166. OrthogonalLine(E_2, i_5) → q_5
  167. Vector(N_2, L_2) → u_5
  168. Vector(N_2, M_2) → v_5
  169. Vector(K_2, F_2) → t_5
  170. Vector(K_2, G_2) → u_6
  171. Polygon(D_2, B_2, C_2, E_2) → poly21, d_9, b_6, c_7, e_9
  172. Rotate(D_2, (-Move), H_2) → I_2
  173. Rotate(B_2, (-Move), H_2) → J_2
  174. Rotate(C_2, (-Move), H_2) → R_2
  175. Rotate(E_2, (-Move), H_2) → S_2
  176. Polygon(I_2, J_2, R_2, S_2) → poly21', i_6, j_6, r_6, s_6
  177. Segment(I_2, S_2) → k_6
  178. OrthogonalLine(I_2, j_6) → l_6
  179. OrthogonalLine(L_2, l_6) → n_6
  180. Intersect(l_6, n_6) → T_2
  181. OrthogonalLine(J_2, n_6) → p_6
  182. Midpoint(L_2, T_2) → V_2
  183. Vector(V_2, L_2) → v_6
  184. Vector(V_2, T_2) → w_6
  185. OrthogonalLine(F_1, f_8) → r_5
  186. OrthogonalLine(G_1, f_8) → s_5
  187. Point(r_5) → O_2
  188. OrthogonalLine(O_2, s_5) → q_6
  189. Intersect(s_5, q_6) → P_2
  190. Midpoint(O_2, P_2) → Q_2
  191. Vector(Q_2, O_2) → w_5
  192. Vector(Q_2, P_2) → t_6
  193. Segment(D_1, E_1) → a_7
  194. Segment(F_1, G_1) → h_7
  195. Segment(F_1, D_1) → b_7
  196. Segment(G_1, E_1) → i_7
  197. Segment(U_2, W_2) → j_7
  198. OrthogonalLine(Z_2, j_7) → k_7
  199. Intersect(k_7, j_7) → A_3
  200. Rotate(Z_2, 180*°, A_3) → B_3
  201. Segment(Z_2, U_2) → l_7
  202. Segment(U_2, B_3) → m_7
  203. Segment(B_3, W_2) → n_7
  204. Segment(W_2, Z_2) → p_7
  205. Line(U_2, W_2) → q_7
  206. Point(q_7) → C_3
  207. Point(k_7) → D_3
  208. Line(C_3, k_7) → r_7
  209. Line(D_3, q_7) → s_7
  210. Intersect(r_7, s_7) → E_3
  211. Line(B_3, q_7) → t_7
  212. Line(W_2, k_7) → a_8
  213. Intersect(s_7, a_8) → F_3
  214. Intersect(r_7, t_7) → G_3
  215. Line(Z_2, q_7) → b_8
  216. Intersect(r_7, b_8) → H_3
  217. Midpoint(H_3, G_3) → I_3
  218. Midpoint(E_3, F_3) → J_3
  219. Vector(J_3, E_3) → u_7
  220. Vector(J_3, F_3) → v_7
  221. Vector(I_3, H_3) → w_7
  222. Vector(I_3, G_3) → u_8
  223. Polygon(U_2, A_3, Z_2) → poly22, z_3, u_9, a_9
  224. Polygon(U_2, B_3, A_3) → poly23, a_{10}, u_{10}, b_9
  225. Polygon(A_3, W_2, B_3) → poly24, b_{10}, a_{11}, w_8
  226. Polygon(A_3, Z_2, W_2) → poly25, w_9, a_{12}, z_4
  227. Mirror(U_2, l_7) → K_3
  228. Mirror(A_3, l_7) → L_3
  229. Mirror(Z_2, l_7) → M_3
  230. Midpoint(Z_2, A_3) → N_3
  231. Midpoint(U_2, Z_2) → O_3
  232. Midpoint(Z_2, W_2) → P_3
  233. Segment(O_3, N_3) → c_8
  234. Segment(N_3, P_3) → g_8
  235. Mirror(U_2, c_8) → Q_3
  236. Mirror(A_3, c_8) → R_3
  237. Mirror(Z_2, c_8) → S_3
  238. Polygon(Q_3, R_3, S_3) → poly22', s_8, q_8, r_8
  239. Mirror(A_3, g_8) → T_3
  240. Mirror(Z_2, g_8) → U_3
  241. Mirror(W_2, g_8) → V_3
  242. Polygon(T_3, U_3, V_3) → poly25', v_8, t_8, u_{11}
  243. Midpoint(U_2, A_3) → W_3
  244. Rotate(Q_3, (-o_5), W_3) → Z_3
  245. Rotate(R_3, (-o_5), W_3) → A_4
  246. Rotate(S_3, (-o_5), W_3) → B_4
  247. Polygon(Z_3, A_4, B_4) → poly22'', b_{11}, z_5, a_{13}
  248. Midpoint(A_3, W_2) → C_4
  249. Rotate(T_3, o_5, C_4) → D_4
  250. Rotate(U_3, o_5, C_4) → E_4
  251. Rotate(V_3, o_5, C_4) → F_4
  252. Polygon(D_4, E_4, F_4) → poly25'', f_9, d_{10}, e_{10}
  253. Segment(U_2, W_2) → h_8
  254. Segment(Z_2, B_3) → i_8
  255. OrthogonalLine(U_2, b_8) → j_8
  256. Intersect(b_8, j_8) → G_4
  257. Intersect(t_7, j_8) → H_4
  258. Intersect(t_7, a_8) → I_4
  259. Intersect(a_8, b_8) → J_4
  260. Polygon(G_4, J_4, I_4, H_4) → poly26, g_9, j_9, i_9, h_9
  261. Segment(A, B) → k_8
  262. Segment(B, C) → l_8
  263. Segment(C, A) → m_8
  264. Angle(J_4, G_4, H_4) → ζ
  265. Angle(H_4, I_4, J_4) → η
  266. Angle(G_4, H_4, I_4) → θ
  267. Angle(I_4, J_4, G_4) → ι
  268. Angle(H_1, D_1, E_1) → κ
  269. Angle(E_1, L_1, H_1) → λ
  270. Angle(L_1, H_1, D_1) → μ
  271. Angle(D_1, E_1, L_1) → ν
  272. Angle(F_1, M_1, Q_1) → ξ
  273. Angle(Q_1, G_1, F_1) → ο
  274. Angle(G_1, F_1, M_1) → ρ
  275. Angle(M_1, Q_1, G_1) → σ
  276. Angle(B, U, O) → τ
  277. Angle(O, C, B) → υ
  278. Angle(C, B, U) → ϕ
  279. Angle(U, O, C) → χ
  280. Angle(B, I, J) → ψ
  281. Angle(I, J, C) → ω
  282. Angle(C, B, I) → α_1