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Circle Area Discovery Activity

Circle Area Discovery Activity
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2022-09-13 20:29
CircleLimitOptimization
This example dynamically demonstrates the discovery of the circle area formula through inscribed polygons. ① Shows how to approximate circle area using polygons; ② Involves core concepts: Circle, Area, Limit, Transformation; ③ Construction uses commands like Circle, Center, Segment, Dilate, Intersect, Rotate, Locus; ④ Suitable for high school mathematics to understand limit concepts and area optimization.

GeoGebra Commands & Parameters

  1. If(2 < a < 3, Translate[h, (((0.9 * pi) * (a - 2)), 0)]) → e_2
  2. Center(e_2) → Q_1
  3. Segment(A, B) → a_1
  4. Dilate(D, If[0 < a < 1, 1 - a, 1], B) → D'
  5. Intersect(c, yAxis, 2) → E
  6. Point(yAxis) → F
  7. Translate(c, If[0 < a < 1, ((pi * a), 0)]) → c'
  8. Center(c') → G
  9. Segment(D, D') → d
  10. Mirror(D, f) → D'_1
  11. Mirror(D', f) → D''
  12. Segment(D'_1, D'') → d'
  13. Intersect(g, f, 2) → J
  14. Mirror(K, f) → K'
  15. Mirror(L, f) → L'
  16. Mirror(N, f) → N'
  17. Mirror(P, f) → P'
  18. Mirror(Q, f) → Q'
  19. Mirror(R, f) → R'
  20. Mirror(S, f) → S'
  21. Mirror(T, f) → T'
  22. Mirror(U, f) → U'
  23. Mirror(V, f) → V'
  24. Mirror(W, f) → W'
  25. Mirror(Z, f) → Z'
  26. Mirror(A_1, f) → M
  27. Mirror(B_1, f) → O
  28. Mirror(C_1, f) → H_1
  29. Mirror(D_1, f) → I_1
  30. Mirror(E_1, f) → J_1
  31. Mirror(F_1, f) → K_1
  32. Mirror(G_1, f) → L_1
  33. Dilate(J, If[1 < a < 2, 2 - a, 1], K) → J'
  34. If(1 < a < 2, Translate[g, (((0.95 * pi) * (a - 1)), 0)]) → t_1
  35. Center(t_1) → N_1
  36. Segment(J, J') → j
  37. Mirror(J', f) → J''_1
  38. Segment(J, J''_1) → b_1
  39. Mirror(J, f) → J'_1
  40. Mirror(J', f) → J''_2
  41. Segment(K', K) → m
  42. Intersect(g, f, 1) → I
  43. Intersect(h, f) → M_1, O_1
  44. Intersect(k, f) → P_1, R_1
  45. Intersect(p, f) → S_1, T_1
  46. If(1 < a < 2, Translate[I, (((0.95 * pi) * (a - 1)), 0)]) → U_1
  47. Rotate(U_1, ((1 - a) * 180°), N_1) → V_1
  48. CircleArc(N_1, J', V_1) → c_2
  49. Mirror(c_2, f) → d_2
  50. If(2 < a < 3, Translate[M_1, (((0.9 * pi) * (a - 2)), 0)]) → W_1
  51. Dilate(O_1, If[2 < a < 3, 3 - a, 1], L) → Z_1
  52. Rotate(W_1, ((2 - a) * 180°), Q_1) → A_2
  53. CircleArc(Q_1, Z_1, A_2) → f_2
  54. Segment(O_1, Z_1) → n
  55. Mirror(f_2, f) → g_2
  56. Mirror(O_1, f) → B_2
  57. Mirror(Z_1, f) → C_2
  58. Segment(B_2, C_2) → n'
  59. Segment(L', L) → j_1
  60. Segment(N', N) → l_1
  61. Intersect(k, f, 1) → D_2
  62. Dilate(R_1, If[3 < a < 4, 4 - a, 1], N) → E_2
  63. If(3 < a < 4, Translate[k, (((0.85 * pi) * (a - 3)), 0)]) → h_2
  64. Center(h_2) → F_2
  65. If(3 < a < 4, Translate[D_2, (((0.85 * pi) * (a - 3)), 0)]) → G_2
  66. Rotate(G_2, ((3 - a) * 180°), F_2) → H_2
  67. CircleArc(F_2, E_2, H_2) → k_2
  68. Segment(R_1, E_2) → m_1
  69. Mirror(k_2, f) → p_2
  70. Mirror(R_1, f) → I_2
  71. Mirror(E_2, f) → J_2
  72. Segment(I_2, J_2) → n_1
  73. Intersect(f, yAxis) → K_2
  74. Intersect(f, yAxis) → L_2
  75. Intersect(f, yAxis) → M_2
  76. Intersect(c, f) → N_2, O_2
  77. If(0 < a < 1, Translate[N_2, ((a * pi), 0)]) → P_2
  78. Rotate(P_2, If[0 < a < 1, ((-a) * 180°)], C) → H
  79. CircleArc(C, D', H) → e
  80. Mirror(e, f) → e'
  81. Intersect(q, f) → Q_2, R_2
  82. Intersect(f, yAxis) → S_2
  83. Intersect(f, yAxis) → T_2
  84. Intersect(f, yAxis) → U_2
  85. Intersect(f, yAxis) → V_2
  86. Intersect(f, yAxis) → W_2
  87. Intersect(f, yAxis) → Z_2
  88. Intersect(f, yAxis) → A_3
  89. Intersect(r, s) → B_3, , ,
  90. Intersect(f, yAxis) → C_3
  91. Intersect(r, f, 1) → D_3
  92. Intersect(s, f, 1) → E_3
  93. Intersect(t, f, 1) → F_3
  94. Intersect(c_1, f, 1) → G_3
  95. Intersect(d_1, f, 1) → H_3
  96. Intersect(e_1, f, 1) → I_3
  97. Intersect(f_1, f, 1) → J_3
  98. Intersect(g_1, f, 1) → K_3
  99. Intersect(h_1, f, 1) → L_3
  100. Intersect(k_1, f, 1) → M_3
  101. Intersect(p_1, f, 1) → N_3
  102. Intersect(q_1, f, 1) → O_3
  103. Intersect(r_1, f, 1) → P_3
  104. Intersect(s_1, f, 1) → Q_3
  105. Intersect(s_1, f, 2) → R_3
  106. Intersect(r_1, f, 2) → S_3
  107. Intersect(q_1, f, 2) → T_3
  108. Intersect(f, yAxis) → U_3
  109. Intersect(p_1, f, 2) → V_3
  110. Intersect(k_1, f, 2) → W_3
  111. Intersect(h_1, f, 2) → Z_3
  112. Intersect(g_1, f, 2) → A_4
  113. Intersect(f_1, f, 2) → B_4
  114. Intersect(e_1, f, 2) → C_4
  115. Intersect(d_1, f, 2) → D_4
  116. Intersect(c_1, f, 2) → E_4
  117. Intersect(t, f, 2) → F_4
  118. Intersect(s, f, 2) → G_4
  119. Intersect(r, f, 2) → H_4
  120. Dilate(T_1, If[4 < a < 5, 5 - a, 1], P) → I_4
  121. Dilate(R_2, If[5 < a < 6, 6 - a, 1], Q) → J_4
  122. Segment(T_1, I_4) → a_2
  123. Mirror(T_1, f) → K_4
  124. Mirror(I_4, f) → L_4
  125. Segment(K_4, L_4) → b_2
  126. Segment(P', P) → i_2
  127. Segment(Q', Q) → j_2
  128. Segment(R', R) → l_2
  129. Segment(S', S) → m_2
  130. Segment(T', T) → n_2
  131. Segment(U', U) → q_2
  132. Segment(V', V) → r_2
  133. Segment(W', W) → s_2
  134. Segment(Z', Z) → t_2
  135. Segment(M, A_1) → a_3
  136. Segment(O, B_1) → b_3
  137. Segment(H_1, C_1) → c_3
  138. Segment(I_1, D_1) → d_3
  139. Segment(J_1, E_1) → e_3
  140. Segment(K_1, F_1) → f_3
  141. Segment(L_1, G_1) → g_3
  142. Segment(R_2, J_4) → h_3
  143. Dilate(H_4, If[6 < a < 7, 7 - a, 1], R) → M_4
  144. Segment(H_4, M_4) → i_3
  145. Dilate(G_4, If[7 < a < 8, 8 - a, 1], S) → N_4
  146. Segment(G_4, N_4) → j_3
  147. Dilate(F_4, If[8 < a < 9, 9 - a, 1], T) → O_4
  148. Segment(F_4, O_4) → k_3
  149. Dilate(E_4, If[9 < a < 10, 10 - a, 1], U) → P_4
  150. Segment(E_4, P_4) → l_3
  151. Dilate(D_4, If[10 < a < 11, 11 - a, 1], V) → Q_4
  152. Segment(D_4, Q_4) → m_3
  153. Dilate(C_4, If[11 < a < 12, 12 - a, 1], W) → R_4
  154. Segment(C_4, R_4) → n_3
  155. Dilate(B_4, If[12 < a < 13, 13 - a, 1], Z) → S_4
  156. Segment(B_4, S_4) → p_3
  157. Dilate(A_4, If[13 < a < 14, 14 - a, 1], A_1) → T_4
  158. Segment(A_4, T_4) → q_3
  159. Dilate(Z_3, If[14 < a < 15, 15 - a, 1], B_1) → U_4
  160. Segment(Z_3, U_4) → r_3
  161. Dilate(W_3, If[15 < a < 16, 16 - a, 1], C_1) → V_4
  162. Segment(W_3, V_4) → s_3
  163. Dilate(V_3, If[16 < a < 17, 17 - a, 1], D_1) → W_4
  164. Segment(V_3, W_4) → t_3
  165. Dilate(T_3, If[17 < a < 18, 18 - a, 1], E_1) → Z_4
  166. Segment(T_3, Z_4) → a_4
  167. Dilate(S_3, If[18 < a < 19, 19 - a, 1], F_1) → A_5
  168. Segment(S_3, A_5) → b_4
  169. Dilate(R_3, If[19 < a < 20, 20 - a, 1], G_1) → B_5
  170. Segment(R_3, B_5) → c_4
  171. If(4 < a < 5, Translate[p, (((0.8 * pi) * (a - 4)), 0)]) → d_4
  172. If(4 < a < 5, Translate[S_1, (((0.8 * pi) * (a - 4)), 0)]) → C_5
  173. Center(d_4) → D_5
  174. Rotate(C_5, ((4 - a) * 180°), D_5) → E_5
  175. CircleArc(D_5, I_4, E_5) → e_4
  176. If(5 < a < 6, Translate[q, (((0.75 * pi) * (a - 5)), 0)]) → f_4
  177. If(5 < a < 6, Translate[Q_2, (((0.75 * pi) * (a - 5)), 0)]) → F_5
  178. Center(f_4) → G_5
  179. Rotate(F_5, ((5 - a) * 180°), G_5) → H_5
  180. CircleArc(G_5, J_4, H_5) → g_4
  181. If(6 < a < 7, Translate[r, (((0.7 * pi) * (a - 6)), 0)]) → h_4
  182. If(6 < a < 7, Translate[D_3, (((0.7 * pi) * (a - 6)), 0)]) → I_5
  183. Center(h_4) → J_5
  184. Rotate(I_5, ((6 - a) * 180°), J_5) → K_5
  185. CircleArc(J_5, M_4, K_5) → k_4
  186. If(7 < a < 8, Translate[s, (((0.65 * pi) * (a - 7)), 0)]) → p_4
  187. If(7 < a < 8, Translate[E_3, (((0.65 * pi) * (a - 7)), 0)]) → L_5
  188. Center(p_4) → M_5
  189. Rotate(L_5, ((7 - a) * 180°), M_5) → N_5
  190. CircleArc(M_5, N_4, N_5) → q_4
  191. If(8 < a < 9, Translate[t, (((0.6 * pi) * (a - 8)), 0)]) → r_4
  192. If(8 < a < 9, Translate[F_3, (((0.6 * pi) * (a - 8)), 0)]) → O_5
  193. Center(r_4) → P_5
  194. Rotate(O_5, ((8 - a) * 180°), P_5) → Q_5
  195. CircleArc(P_5, O_4, Q_5) → s_4
  196. If(9 < a < 10, Translate[c_1, (((0.55 * pi) * (a - 9)), 0)]) → t_4
  197. If(9 < a < 10, Translate[G_3, (((0.55 * pi) * (a - 9)), 0)]) → R_5
  198. Center(t_4) → S_5
  199. Rotate(R_5, ((9 - a) * 180°), S_5) → T_5
  200. CircleArc(S_5, P_4, T_5) → c_5
  201. If(10 < a < 11, Translate[d_1, (((0.5 * pi) * (a - 10)), 0)]) → d_5
  202. If(10 < a < 11, Translate[H_3, (((0.5 * pi) * (a - 10)), 0)]) → U_5
  203. Center(d_5) → V_5
  204. Rotate(U_5, ((10 - a) * 180°), V_5) → W_5
  205. CircleArc(V_5, Q_4, W_5) → e_5
  206. If(11 < a < 12, Translate[e_1, (((0.45 * pi) * (a - 11)), 0)]) → f_5
  207. If(11 < a < 12, Translate[I_3, (((0.45 * pi) * (a - 11)), 0)]) → Z_5
  208. Center(f_5) → A_6
  209. Rotate(Z_5, ((11 - a) * 180°), A_6) → B_6
  210. CircleArc(A_6, R_4, B_6) → g_5
  211. If(12 < a < 13, Translate[f_1, (((0.4 * pi) * (a - 12)), 0)]) → h_5
  212. If(12 < a < 13, Translate[J_3, (((0.4 * pi) * (a - 12)), 0)]) → C_6
  213. Center(h_5) → D_6
  214. Rotate(C_6, ((12 - a) * 180°), D_6) → E_6
  215. CircleArc(D_6, S_4, E_6) → k_5
  216. If(13 < a < 14, Translate[g_1, (((0.35 * pi) * (a - 13)), 0)]) → p_5
  217. If(13 < a < 14, Translate[K_3, (((0.35 * pi) * (a - 13)), 0)]) → F_6
  218. Center(p_5) → G_6
  219. Rotate(F_6, ((13 - a) * 180°), G_6) → H_6
  220. CircleArc(G_6, T_4, H_6) → q_5
  221. If(14 < a < 15, Translate[h_1, (((0.3 * pi) * (a - 14)), 0)]) → r_5
  222. If(14 < a < 15, Translate[L_3, (((0.3 * pi) * (a - 14)), 0)]) → I_6
  223. Center(r_5) → J_6
  224. Rotate(I_6, ((14 - a) * 180°), J_6) → K_6
  225. CircleArc(J_6, U_4, K_6) → s_5
  226. If(15 < a < 16, Translate[k_1, (((0.25 * pi) * (a - 15)), 0)]) → t_5
  227. If(15 < a < 16, Translate[M_3, (((0.25 * pi) * (a - 15)), 0)]) → L_6
  228. Center(t_5) → M_6
  229. Rotate(L_6, ((15 - a) * 180°), M_6) → N_6
  230. CircleArc(M_6, V_4, N_6) → c_6
  231. If(16 < a < 17, Translate[p_1, (((0.2 * pi) * (a - 16)), 0)]) → d_6
  232. If(16 < a < 17, Translate[N_3, (((0.2 * pi) * (a - 16)), 0)]) → O_6
  233. Center(d_6) → P_6
  234. Rotate(O_6, ((16 - a) * 180°), P_6) → Q_6
  235. CircleArc(P_6, W_4, Q_6) → e_6
  236. If(17 < a < 18, Translate[q_1, (((0.15 * pi) * (a - 17)), 0)]) → f_6
  237. If(17 < a < 18, Translate[O_3, (((0.15 * pi) * (a - 17)), 0)]) → R_6
  238. Center(f_6) → S_6
  239. Rotate(R_6, ((17 - a) * 180°), S_6) → T_6
  240. CircleArc(S_6, Z_4, T_6) → g_6
  241. If(18 < a < 19, Translate[r_1, (((0.1 * pi) * (a - 18)), 0)]) → h_6
  242. If(18 < a < 19, Translate[P_3, (((0.1 * pi) * (a - 18)), 0)]) → U_6
  243. Center(h_6) → V_6
  244. Rotate(U_6, ((18 - a) * 180°), V_6) → W_6
  245. CircleArc(V_6, A_5, W_6) → k_6
  246. If(19 < a < 20, Translate[s_1, (((0.05 * pi) * (a - 19)), 0)]) → p_6
  247. If(19 < a < 20, Translate[Q_3, (((0.05 * pi) * (a - 19)), 0)]) → Z_6
  248. Center(p_6) → A_7
  249. Rotate(Z_6, ((19 - a) * 180°), A_7) → B_7
  250. CircleArc(A_7, B_5, B_7) → q_6
  251. Mirror(e_4, f) → r_6
  252. Mirror(R_2, f) → C_7
  253. Mirror(J_4, f) → D_7
  254. Segment(C_7, D_7) → i_4
  255. Mirror(g_4, f) → s_6
  256. Mirror(k_4, f) → t_6
  257. Mirror(H_4, f) → E_7
  258. Mirror(M_4, f) → F_7
  259. Segment(E_7, F_7) → j_4
  260. Mirror(G_4, f) → G_7
  261. Mirror(N_4, f) → H_7
  262. Segment(G_7, H_7) → l_4
  263. Mirror(q_4, f) → c_7
  264. Mirror(F_4, f) → I_7
  265. Mirror(O_4, f) → J_7
  266. Segment(I_7, J_7) → m_4
  267. Mirror(s_4, f) → d_7
  268. Mirror(E_4, f) → K_7
  269. Mirror(P_4, f) → L_7
  270. Segment(K_7, L_7) → n_4
  271. Mirror(c_5, f) → e_7
  272. Mirror(D_4, f) → M_7
  273. Mirror(Q_4, f) → N_7
  274. Segment(M_7, N_7) → a_5
  275. Mirror(e_5, f) → f_7
  276. Mirror(C_4, f) → O_7
  277. Mirror(R_4, f) → P_7
  278. Segment(O_7, P_7) → b_5
  279. Mirror(g_5, f) → g_7
  280. Mirror(B_4, f) → Q_7
  281. Mirror(S_4, f) → R_7
  282. Segment(Q_7, R_7) → i_5
  283. Mirror(k_5, f) → h_7
  284. Mirror(A_4, f) → S_7
  285. Mirror(T_4, f) → T_7
  286. Segment(S_7, T_7) → j_5
  287. Mirror(q_5, f) → k_7
  288. Mirror(Z_3, f) → U_7
  289. Mirror(U_4, f) → V_7
  290. Segment(U_7, V_7) → l_5
  291. Mirror(s_5, f) → p_7
  292. Mirror(W_3, f) → W_7
  293. Mirror(V_4, f) → Z_7
  294. Segment(W_7, Z_7) → m_5
  295. Mirror(c_6, f) → q_7
  296. Mirror(V_3, f) → A_8
  297. Mirror(W_4, f) → B_8
  298. Segment(A_8, B_8) → n_5
  299. Mirror(e_6, f) → r_7
  300. Mirror(T_3, f) → C_8
  301. Mirror(Z_4, f) → D_8
  302. Segment(C_8, D_8) → a_6
  303. Mirror(g_6, f) → s_7
  304. Mirror(S_3, f) → E_8
  305. Mirror(A_5, f) → F_8
  306. Segment(E_8, F_8) → b_6
  307. Mirror(k_6, f) → t_7
  308. Mirror(R_3, f) → G_8
  309. Mirror(B_5, f) → H_8
  310. Segment(G_8, H_8) → i_6
  311. Mirror(q_6, f) → c_8
  312. Segment((0, 0), (0, 1)) → i
  313. Segment(A, F) → j_6
  314. Segment(F, B) → l_6
  315. Circle((0, -2), 1) → e_8
  316. Segment((0, -2), (0, -3)) → a_7

AI Drawing Prompt Reference

Draw a circle and approximate its area using inscribed polygons. Start by creating a circle centered at the origin, then use rotation and scaling operations to gradually increase the number of polygon sides, observing how the polygon's area approaches the circle's area. Finally, show the circle's area formula.
The prompt above was compiled from the original drawing commands and is for reference only.