MathPlotAI Logo MathPlotAI

Circle Area Dissection

Circle Area Dissection
Loading the math board and drawing, please wait…
2013-08-03 04:30
CircleTrigonometryTransformationGeometric ConstructionSlider Animation
In collections Area Formulas Area Formulas
This example visually demonstrates the derivation of the circle area formula by dissecting and rearranging circular segments. It involves core concepts like circles, equilateral triangles, quadrilaterals, geometric transformations, trapezoids, and multiplication. GeoGebra commands and objects used include Circle, Center, Intersect, Rotate, Segment, Angle, CircleSector, Midpoint, and Ray. Suitable for high school mathematics education to help students grasp the geometric meaning of the circle area formula.

GeoGebra Commands & Parameters

  1. Circle((0, 2), 2) → d
  2. Center(d) → B
  3. Intersect(xAxis, yAxis) → C
  4. Intersect(d, xAxis) → A, E
  5. Rotate(A1, (360 * ° / a), B) → A2
  6. Rotate(A2, (360 * ° / a), B) → A3
  7. Rotate(A3, (360 * ° / a), B) → A4
  8. Rotate(A4, (360 * ° / a), B) → A5
  9. Rotate(A5, (360 * ° / a), B) → A6
  10. Rotate(A6, (360 * ° / a), B) → A7
  11. Rotate(A7, (360 * ° / a), B) → A8
  12. Rotate(A8, (360 * ° / a), B) → A9
  13. Rotate(A9, (360 * ° / a), B) → A10
  14. Rotate(A10, (360 * ° / a), B) → A11
  15. Rotate(A11, (360 * ° / a), B) → A12
  16. Rotate(A12, (360 * ° / a), B) → A13
  17. Rotate(A13, (360 * ° / a), B) → A14
  18. Rotate(A14, (360 * ° / a), B) → A15
  19. Rotate(A15, (360 * ° / a), B) → A16
  20. Rotate(A16, (360 * ° / a), B) → A17
  21. Rotate(A17, (360 * ° / a), B) → A18
  22. Rotate(A18, (360 * ° / a), B) → A19
  23. Rotate(A19, (360 * ° / a), B) → A20
  24. Rotate(d, (-α), A) → d'
  25. Center(d') → F
  26. Rotate(A2, (-α), A) → D2
  27. Rotate(A3, (-α), A) → D3
  28. Rotate(A4, (-α), A) → D4
  29. Rotate(A5, (-α), A) → D5
  30. Rotate(A6, (-α), A) → D6
  31. Rotate(A7, (-α), A) → D7
  32. Rotate(A8, (-α), A) → D8
  33. Rotate(A9, (-α), A) → D9
  34. Rotate(A10, (-α), A) → D10
  35. Rotate(A11, (-α), A) → D11
  36. Rotate(A12, (-α), A) → D12
  37. Rotate(A13, (-α), A) → D13
  38. Rotate(A14, (-α), A) → D14
  39. Rotate(A15, (-α), A) → D15
  40. Rotate(A16, (-α), A) → D16
  41. Rotate(A17, (-α), A) → D17
  42. Rotate(A18, (-α), A) → D18
  43. Rotate(A19, (-α), A) → D19
  44. Rotate(A20, (-α), A) → D20
  45. Segment(A, A2) → B1
  46. Angle(A2, A, C1) → β
  47. Rotate(B, (-β), A) → B'
  48. CircleSector(B', A, C1) → f
  49. Circle(B', C1) → g
  50. Midpoint(C1, C2) → F2
  51. Midpoint(C2, C3) → F3
  52. Midpoint(C3, C4) → F4
  53. Midpoint(C4, C5) → F5
  54. Midpoint(C5, C6) → F6
  55. Midpoint(C6, C7) → F7
  56. Midpoint(C7, C8) → F8
  57. Midpoint(C8, C9) → F9
  58. Midpoint(C9, C10) → F10
  59. Midpoint(C10, C11) → F11
  60. Midpoint(C11, C12) → F12
  61. Midpoint(C12, C13) → F13
  62. Midpoint(C13, C14) → F14
  63. Midpoint(C14, C15) → F15
  64. Midpoint(C15, C16) → F16
  65. Midpoint(C16, C17) → F17
  66. Midpoint(C17, C18) → F18
  67. Midpoint(C18, C19) → F19
  68. Midpoint(C19, C20) → F20
  69. Circle(G2, C2) → h
  70. Circle(G3, C3) → k
  71. Circle(G4, C4) → p
  72. Circle(G5, C5) → q
  73. Circle(G6, C6) → r
  74. Circle(G7, C7) → s
  75. Circle(G8, C8) → t
  76. Circle(G9, C9) → c_1
  77. Circle(G10, C10) → d_1
  78. Circle(G11, C11) → e_1
  79. Circle(G12, C12) → f_1
  80. Circle(G13, C13) → g_1
  81. Circle(G14, C14) → h_1
  82. Circle(G15, C15) → k_1
  83. Circle(G16, C16) → p_1
  84. Circle(G17, C17) → q_1
  85. Circle(G18, C18) → r_1
  86. Circle(G19, C19) → s_1
  87. Circle(G20, C20) → t_1
  88. Rotate(g, (-(α - β)), C1) → g'
  89. CircleSector(G2, C1, C2) → c_2
  90. CircleSector(G3, C2, C3) → d_2
  91. Rotate(h, (-(α - (3 * β))), C2) → h'
  92. CircleSector(F, A1, D2) → E1
  93. CircleSector(F, D2, D3) → E2
  94. CircleSector(F, D3, D4) → E3
  95. CircleSector(F, D4, D5) → E4
  96. CircleSector(F, D5, D6) → E5
  97. CircleSector(F, D6, D7) → E6
  98. CircleSector(F, D7, D8) → E7
  99. CircleSector(F, D8, D9) → E8
  100. CircleSector(F, D9, D10) → E9
  101. CircleSector(F, D10, D11) → E10
  102. CircleSector(F, D11, D12) → E11
  103. CircleSector(F, D12, D13) → E12
  104. CircleSector(F, D13, D14) → E13
  105. CircleSector(F, D14, D15) → E14
  106. CircleSector(F, D15, D16) → E15
  107. CircleSector(F, D16, D17) → E16
  108. CircleSector(F, D17, D18) → E17
  109. CircleSector(F, D18, D19) → E18
  110. CircleSector(F, D19, D20) → E19
  111. CircleSector(F, D20, A1) → E20
  112. CircleSector(G4, C3, C4) → e
  113. CircleSector(G6, C5, C6) → I6
  114. CircleSector(G7, C6, C7) → I7
  115. CircleSector(G8, C7, C8) → I8
  116. CircleSector(G9, C8, C9) → I9
  117. CircleSector(G10, C9, C10) → I10
  118. CircleSector(G11, C10, C11) → I11
  119. CircleSector(G12, C11, C12) → I12
  120. CircleSector(G13, C12, C13) → I13
  121. CircleSector(G14, C13, C14) → I14
  122. CircleSector(G15, C14, C15) → I15
  123. CircleSector(G16, C15, C16) → I16
  124. CircleSector(G17, C16, C17) → I17
  125. CircleSector(G18, C17, C18) → I18
  126. CircleSector(G19, C18, C19) → I19
  127. CircleSector(G20, C19, C20) → I20
  128. Center(g') → D
  129. Rotate(C1, (360 * ° / a), D) → J2
  130. Rotate(J2, (360 * ° / a), D) → J3
  131. Rotate(J3, (360 * ° / a), D) → J4
  132. Rotate(J4, (360 * ° / a), D) → J5
  133. Rotate(J5, (360 * ° / a), D) → J6
  134. Rotate(J6, (360 * ° / a), D) → J7
  135. Rotate(J7, (360 * ° / a), D) → J8
  136. Rotate(J8, (360 * ° / a), D) → J9
  137. Rotate(J9, (360 * ° / a), D) → J10
  138. Rotate(J10, (360 * ° / a), D) → J11
  139. Rotate(J11, (360 * ° / a), D) → J12
  140. Rotate(J12, (360 * ° / a), D) → J13
  141. Rotate(J13, (360 * ° / a), D) → J14
  142. Rotate(J14, (360 * ° / a), D) → J15
  143. Rotate(J15, (360 * ° / a), D) → J16
  144. Rotate(J16, (360 * ° / a), D) → J17
  145. Rotate(J17, (360 * ° / a), D) → J18
  146. Rotate(J18, (360 * ° / a), D) → J19
  147. Rotate(J19, (360 * ° / a), D) → J20
  148. CircleSector(G5, C4, C5) → I5
  149. CircleSector(D, C1, J2) → K2
  150. CircleSector(D, J3, J4) → K4
  151. CircleSector(D, J4, J5) → K5
  152. CircleSector(D, J5, J6) → K6
  153. CircleSector(D, J6, J7) → K7
  154. CircleSector(D, J7, J8) → K8
  155. CircleSector(D, J8, J9) → K9
  156. CircleSector(D, J9, J10) → K10
  157. CircleSector(D, J10, J11) → K11
  158. CircleSector(D, J11, J12) → K12
  159. CircleSector(D, J12, J13) → K13
  160. CircleSector(D, J13, J14) → K14
  161. CircleSector(D, J14, J15) → K15
  162. CircleSector(D, J15, J16) → K16
  163. CircleSector(D, J16, J17) → K17
  164. CircleSector(D, J17, J18) → K18
  165. CircleSector(D, J18, J19) → K19
  166. CircleSector(D, J19, J20) → K20
  167. Center(h') → G
  168. Rotate(C2, (360 * ° / a), G) → L3
  169. Rotate(L3, (360 * ° / a), G) → L4
  170. Rotate(L4, (360 * ° / a), G) → L5
  171. Rotate(L5, (360 * ° / a), G) → L6
  172. Rotate(L6, (360 * ° / a), G) → L7
  173. Rotate(L7, (360 * ° / a), G) → L8
  174. Rotate(L8, (360 * ° / a), G) → L9
  175. Rotate(L9, (360 * ° / a), G) → L10
  176. Rotate(L10, (360 * ° / a), G) → L11
  177. Rotate(L11, (360 * ° / a), G) → L12
  178. Rotate(L12, (360 * ° / a), G) → L13
  179. Rotate(L13, (360 * ° / a), G) → L14
  180. Rotate(L14, (360 * ° / a), G) → L15
  181. Rotate(L15, (360 * ° / a), G) → L16
  182. Rotate(L16, (360 * ° / a), G) → L17
  183. Rotate(L17, (360 * ° / a), G) → L18
  184. Rotate(L18, (360 * ° / a), G) → L19
  185. Rotate(L19, (360 * ° / a), G) → L20
  186. CircleSector(D, J2, J3) → K3
  187. CircleSector(G, C2, L3) → M3
  188. CircleSector(G, L3, L4) → M4
  189. CircleSector(G, L4, L5) → M5
  190. CircleSector(G, L5, L6) → M6
  191. CircleSector(G, L6, L7) → M7
  192. CircleSector(G, L7, L8) → M8
  193. CircleSector(G, L8, L9) → M9
  194. CircleSector(G, L9, L10) → M10
  195. CircleSector(G, L10, L11) → M11
  196. CircleSector(G, L11, L12) → M12
  197. CircleSector(G, L12, L13) → M13
  198. CircleSector(G, L13, L14) → M14
  199. CircleSector(G, L14, L15) → M15
  200. CircleSector(G, L15, L16) → M16
  201. CircleSector(G, L16, L17) → M17
  202. CircleSector(G, L17, L18) → M18
  203. CircleSector(G, L18, L19) → M19
  204. CircleSector(G, L19, L20) → M20
  205. Rotate(k, (-(α - (5 * β))), C3) → k'
  206. Center(k') → H
  207. Rotate(C3, (360 * ° / a), H) → N4
  208. Rotate(N4, (360 * ° / a), H) → N5
  209. Rotate(N5, (360 * ° / a), H) → N6
  210. Rotate(N6, (360 * ° / a), H) → N7
  211. Rotate(N7, (360 * ° / a), H) → N8
  212. Rotate(N8, (360 * ° / a), H) → N9
  213. Rotate(N9, (360 * ° / a), H) → N10
  214. Rotate(N10, (360 * ° / a), H) → N11
  215. Rotate(N11, (360 * ° / a), H) → N12
  216. Rotate(N12, (360 * ° / a), H) → N13
  217. Rotate(N13, (360 * ° / a), H) → N14
  218. Rotate(N14, (360 * ° / a), H) → N15
  219. Rotate(N15, (360 * ° / a), H) → N16
  220. Rotate(N16, (360 * ° / a), H) → N17
  221. Rotate(N17, (360 * ° / a), H) → N18
  222. Rotate(N18, (360 * ° / a), H) → N19
  223. Rotate(N19, (360 * ° / a), H) → N20
  224. CircleSector(H, C3, N4) → O4
  225. CircleSector(H, N4, N5) → O5
  226. CircleSector(H, N5, N6) → O6
  227. CircleSector(H, N6, N7) → O7
  228. CircleSector(H, N7, N8) → O8
  229. CircleSector(H, N8, N9) → O9
  230. CircleSector(H, N9, N10) → O10
  231. CircleSector(H, N10, N11) → O11
  232. CircleSector(H, N11, N12) → O12
  233. CircleSector(H, N12, N13) → O13
  234. CircleSector(H, N13, N14) → O14
  235. CircleSector(H, N14, N15) → O15
  236. CircleSector(H, N15, N16) → O16
  237. CircleSector(H, N16, N17) → O17
  238. CircleSector(H, N17, N18) → O18
  239. CircleSector(H, N18, N19) → O19
  240. CircleSector(H, N19, N20) → O20
  241. Rotate(p, (-(α - (7 * β))), C4) → p'
  242. Center(p') → I
  243. Rotate(C4, (360 * ° / a), I) → P5
  244. Rotate(P5, (360 * ° / a), I) → P6
  245. Rotate(P6, (360 * ° / a), I) → P7
  246. Rotate(P7, (360 * ° / a), I) → P8
  247. Rotate(P8, (360 * ° / a), I) → P9
  248. Rotate(P9, (360 * ° / a), I) → P10
  249. Rotate(P10, (360 * ° / a), I) → P11
  250. Rotate(P11, (360 * ° / a), I) → P12
  251. Rotate(P12, (360 * ° / a), I) → P13
  252. Rotate(P13, (360 * ° / a), I) → P14
  253. Rotate(P14, (360 * ° / a), I) → P15
  254. Rotate(P15, (360 * ° / a), I) → P16
  255. Rotate(P16, (360 * ° / a), I) → P17
  256. Rotate(P17, (360 * ° / a), I) → P18
  257. Rotate(P18, (360 * ° / a), I) → P19
  258. Rotate(P19, (360 * ° / a), I) → P20
  259. CircleSector(I, C4, P5) → Q5
  260. CircleSector(I, P5, P6) → Q6
  261. CircleSector(I, P6, P7) → Q7
  262. CircleSector(I, P7, P8) → Q8
  263. CircleSector(I, P8, P9) → Q9
  264. CircleSector(I, P9, P10) → Q10
  265. CircleSector(I, P10, P11) → Q11
  266. CircleSector(I, P11, P12) → Q12
  267. CircleSector(I, P12, P13) → Q13
  268. CircleSector(I, P13, P14) → Q14
  269. CircleSector(I, P14, P15) → Q15
  270. CircleSector(I, P15, P16) → Q16
  271. CircleSector(I, P16, P17) → Q17
  272. CircleSector(I, P17, P18) → Q18
  273. CircleSector(I, P18, P19) → Q19
  274. CircleSector(I, P19, P20) → Q20
  275. Rotate(q, (-(α - (9 * β))), C5) → q'
  276. Center(q') → J
  277. Rotate(C5, (360 * ° / a), J) → R6
  278. Rotate(R6, (360 * ° / a), J) → R7
  279. Rotate(R7, (360 * ° / a), J) → R8
  280. Rotate(R8, (360 * ° / a), J) → R9
  281. Rotate(R9, (360 * ° / a), J) → R10
  282. Rotate(R10, (360 * ° / a), J) → R11
  283. Rotate(R11, (360 * ° / a), J) → R12
  284. Rotate(R12, (360 * ° / a), J) → R13
  285. Rotate(R13, (360 * ° / a), J) → R14
  286. Rotate(R14, (360 * ° / a), J) → R15
  287. Rotate(R15, (360 * ° / a), J) → R16
  288. Rotate(R16, (360 * ° / a), J) → R17
  289. Rotate(R17, (360 * ° / a), J) → R18
  290. Rotate(R18, (360 * ° / a), J) → R19
  291. Rotate(R19, (360 * ° / a), J) → R20
  292. CircleSector(J, C5, R6) → S6
  293. CircleSector(J, R6, R7) → S7
  294. CircleSector(J, R7, R8) → S8
  295. CircleSector(J, R8, R9) → S9
  296. CircleSector(J, R9, R10) → S10
  297. CircleSector(J, R10, R11) → S11
  298. CircleSector(J, R11, R12) → S12
  299. CircleSector(J, R12, R13) → S13
  300. CircleSector(J, R13, R14) → S14
  301. CircleSector(J, R14, R15) → S15
  302. CircleSector(J, R15, R16) → S16
  303. CircleSector(J, R16, R17) → S17
  304. CircleSector(J, R17, R18) → S18
  305. CircleSector(J, R18, R19) → S19
  306. CircleSector(J, R19, R20) → S20
  307. Rotate(r, (-(α - (11 * β))), C6) → r'
  308. Center(r') → K
  309. Rotate(C6, (360 * ° / a), K) → T7
  310. Rotate(T7, (360 * ° / a), K) → T8
  311. Rotate(T8, (360 * ° / a), K) → T9
  312. Rotate(T9, (360 * ° / a), K) → T10
  313. Rotate(T10, (360 * ° / a), K) → T11
  314. Rotate(T11, (360 * ° / a), K) → T12
  315. Rotate(T12, (360 * ° / a), K) → T13
  316. Rotate(T13, (360 * ° / a), K) → T14
  317. Rotate(T14, (360 * ° / a), K) → T15
  318. Rotate(T15, (360 * ° / a), K) → T16
  319. Rotate(T16, (360 * ° / a), K) → T17
  320. Rotate(T17, (360 * ° / a), K) → T18
  321. Rotate(T18, (360 * ° / a), K) → T19
  322. Rotate(T19, (360 * ° / a), K) → T20
  323. CircleSector(K, C6, T7) → U7
  324. CircleSector(K, T7, T8) → U8
  325. CircleSector(K, T8, T9) → U9
  326. CircleSector(K, T9, T10) → U10
  327. CircleSector(K, T10, T11) → U11
  328. CircleSector(K, T11, T12) → U12
  329. CircleSector(K, T12, T13) → U13
  330. CircleSector(K, T13, T14) → U14
  331. CircleSector(K, T14, T15) → U15
  332. CircleSector(K, T15, T16) → U16
  333. CircleSector(K, T16, T17) → U17
  334. CircleSector(K, T17, T18) → U18
  335. CircleSector(K, T18, T19) → U19
  336. CircleSector(K, T19, T20) → U20
  337. Rotate(s, (-(α - (13 * β))), C7) → s'
  338. Center(s') → L
  339. Rotate(C7, (360 * ° / a), L) → V8
  340. Rotate(V8, (360 * ° / a), L) → V9
  341. Rotate(V9, (360 * ° / a), L) → V10
  342. Rotate(V10, (360 * ° / a), L) → V11
  343. Rotate(V11, (360 * ° / a), L) → V12
  344. Rotate(V12, (360 * ° / a), L) → V13
  345. Rotate(V13, (360 * ° / a), L) → V14
  346. Rotate(V14, (360 * ° / a), L) → V15
  347. Rotate(V15, (360 * ° / a), L) → V16
  348. Rotate(V16, (360 * ° / a), L) → V17
  349. Rotate(V17, (360 * ° / a), L) → V18
  350. Rotate(V18, (360 * ° / a), L) → V19
  351. Rotate(V19, (360 * ° / a), L) → V20
  352. CircleSector(L, C7, V8) → W8
  353. CircleSector(L, V8, V9) → W9
  354. CircleSector(L, V9, V10) → W10
  355. CircleSector(L, V10, V11) → W11
  356. CircleSector(L, V11, V12) → W12
  357. CircleSector(L, V12, V13) → W13
  358. CircleSector(L, V13, V14) → W14
  359. CircleSector(L, V14, V15) → W15
  360. CircleSector(L, V15, V16) → W16
  361. CircleSector(L, V16, V17) → W17
  362. CircleSector(L, V17, V18) → W18
  363. CircleSector(L, V18, V19) → W19
  364. CircleSector(L, V19, V20) → W20
  365. Rotate(t, (-(α - (15 * β))), C8) → t'
  366. Center(t') → M
  367. Rotate(C8, (360 * ° / a), M) → X9
  368. Rotate(X9, (360 * ° / a), M) → X10
  369. Rotate(X10, (360 * ° / a), M) → X11
  370. Rotate(X11, (360 * ° / a), M) → X12
  371. Rotate(X12, (360 * ° / a), M) → X13
  372. Rotate(X13, (360 * ° / a), M) → X14
  373. Rotate(X14, (360 * ° / a), M) → X15
  374. Rotate(X15, (360 * ° / a), M) → X16
  375. Rotate(X16, (360 * ° / a), M) → X17
  376. Rotate(X17, (360 * ° / a), M) → X18
  377. Rotate(X18, (360 * ° / a), M) → X19
  378. Rotate(X19, (360 * ° / a), M) → X20
  379. CircleSector(M, C8, X9) → Y9
  380. CircleSector(M, X9, X10) → Y10
  381. CircleSector(M, X10, X11) → Y11
  382. CircleSector(M, X11, X12) → Y12
  383. CircleSector(M, X12, X13) → Y13
  384. CircleSector(M, X13, X14) → Y14
  385. CircleSector(M, X14, X15) → Y15
  386. CircleSector(M, X15, X16) → Y16
  387. CircleSector(M, X16, X17) → Y17
  388. CircleSector(M, X17, X18) → Y18
  389. CircleSector(M, X18, X19) → Y19
  390. CircleSector(M, X19, X20) → Y20
  391. Rotate(c_1, (-(α - (17 * β))), C9) → e_2
  392. Rotate(d_1, (-(α - (19 * β))), C10) → f_2
  393. Center(e_2) → N
  394. Rotate(C9, (360 * ° / a), N) → Z10
  395. Rotate(Z10, (360 * ° / a), N) → Z11
  396. Rotate(Z11, (360 * ° / a), N) → Z12
  397. Rotate(Z12, (360 * ° / a), N) → Z13
  398. Rotate(Z13, (360 * ° / a), N) → Z14
  399. Rotate(Z14, (360 * ° / a), N) → Z15
  400. Rotate(Z15, (360 * ° / a), N) → Z16
  401. Rotate(Z16, (360 * ° / a), N) → Z17
  402. Rotate(Z17, (360 * ° / a), N) → Z18
  403. Rotate(Z18, (360 * ° / a), N) → Z19
  404. Rotate(Z19, (360 * ° / a), N) → Z20
  405. CircleSector(N, C9, Z10) → AA10
  406. CircleSector(N, Z10, Z11) → AA11
  407. CircleSector(N, Z11, Z12) → AA12
  408. CircleSector(N, Z12, Z13) → AA13
  409. CircleSector(N, Z13, Z14) → AA14
  410. CircleSector(N, Z14, Z15) → AA15
  411. CircleSector(N, Z15, Z16) → AA16
  412. CircleSector(N, Z16, Z17) → AA17
  413. CircleSector(N, Z17, Z18) → AA18
  414. CircleSector(N, Z18, Z19) → AA19
  415. CircleSector(N, Z19, Z20) → AA20
  416. Center(f_2) → O
  417. Rotate(C10, (360 * ° / a), O) → AB11
  418. Rotate(AB11, (360 * ° / a), O) → AB12
  419. Rotate(AB12, (360 * ° / a), O) → AB13
  420. Rotate(AB13, (360 * ° / a), O) → AB14
  421. Rotate(AB14, (360 * ° / a), O) → AB15
  422. Rotate(AB15, (360 * ° / a), O) → AB16
  423. Rotate(AB16, (360 * ° / a), O) → AB17
  424. Rotate(AB17, (360 * ° / a), O) → AB18
  425. Rotate(AB18, (360 * ° / a), O) → AB19
  426. Rotate(AB19, (360 * ° / a), O) → AB20
  427. CircleSector(O, C10, AB11) → AC11
  428. CircleSector(O, AB11, AB12) → AC12
  429. CircleSector(O, AB12, AB13) → AC13
  430. CircleSector(O, AB13, AB14) → AC14
  431. CircleSector(O, AB14, AB15) → AC15
  432. CircleSector(O, AB15, AB16) → AC16
  433. CircleSector(O, AB16, AB17) → AC17
  434. CircleSector(O, AB17, AB18) → AC18
  435. CircleSector(O, AB18, AB19) → AC19
  436. CircleSector(O, AB19, AB20) → AC20
  437. Rotate(e_1, (-(α - (21 * β))), C11) → g_2
  438. Center(g_2) → P
  439. Rotate(C11, (360 * ° / a), P) → AD12
  440. Rotate(AD12, (360 * ° / a), P) → AD13
  441. Rotate(AD13, (360 * ° / a), P) → AD14
  442. Rotate(AD14, (360 * ° / a), P) → AD15
  443. Rotate(AD15, (360 * ° / a), P) → AD16
  444. Rotate(AD16, (360 * ° / a), P) → AD17
  445. Rotate(AD17, (360 * ° / a), P) → AD18
  446. Rotate(AD18, (360 * ° / a), P) → AD19
  447. Rotate(AD19, (360 * ° / a), P) → AD20
  448. CircleSector(P, C11, AD12) → AE12
  449. CircleSector(P, AD12, AD13) → AE13
  450. CircleSector(P, AD13, AD14) → AE14
  451. CircleSector(P, AD14, AD15) → AE15
  452. CircleSector(P, AD15, AD16) → AE16
  453. CircleSector(P, AD16, AD17) → AE17
  454. CircleSector(P, AD17, AD18) → AE18
  455. CircleSector(P, AD18, AD19) → AE19
  456. CircleSector(P, AD19, AD20) → AE20
  457. Rotate(f_1, (-(α - (23 * β))), C12) → h_2
  458. Center(h_2) → Q
  459. Rotate(C12, (360 * ° / a), Q) → AF13
  460. Rotate(AF13, (360 * ° / a), Q) → AF14
  461. Rotate(AF14, (360 * ° / a), Q) → AF15
  462. Rotate(AF15, (360 * ° / a), Q) → AF16
  463. Rotate(AF16, (360 * ° / a), Q) → AF17
  464. Rotate(AF17, (360 * ° / a), Q) → AF18
  465. Rotate(AF18, (360 * ° / a), Q) → AF19
  466. Rotate(AF19, (360 * ° / a), Q) → AF20
  467. CircleSector(Q, C12, AF13) → AG13
  468. CircleSector(Q, AF13, AF14) → AG14
  469. CircleSector(Q, AF14, AF15) → AG15
  470. CircleSector(Q, AF15, AF16) → AG16
  471. CircleSector(Q, AF16, AF17) → AG17
  472. CircleSector(Q, AF17, AF18) → AG18
  473. CircleSector(Q, AF18, AF19) → AG19
  474. CircleSector(Q, AF19, AF20) → AG20
  475. Rotate(g_1, (-(α - (25 * β))), C13) → k_2
  476. Center(k_2) → R
  477. Center(k_2) → S
  478. Rotate(C13, (360 * ° / a), S) → AH14
  479. Rotate(AH14, (360 * ° / a), S) → AH15
  480. Rotate(AH15, (360 * ° / a), S) → AH16
  481. Rotate(AH16, (360 * ° / a), S) → AH17
  482. Rotate(AH17, (360 * ° / a), S) → AH18
  483. Rotate(AH18, (360 * ° / a), S) → AH19
  484. Rotate(AH19, (360 * ° / a), S) → AH20
  485. CircleSector(S, C13, AH14) → AI14
  486. CircleSector(S, AH14, AH15) → AI15
  487. CircleSector(S, AH15, AH16) → AI16
  488. CircleSector(S, AH16, AH17) → AI17
  489. CircleSector(S, AH17, AH18) → AI18
  490. CircleSector(S, AH18, AH19) → AI19
  491. CircleSector(S, AH19, AH20) → AI20
  492. Rotate(h_1, (-(α - (27 * β))), C14) → p_2
  493. Center(p_2) → T
  494. Rotate(C14, (360 * ° / a), T) → AJ15
  495. Rotate(AJ15, (360 * ° / a), T) → AJ16
  496. Rotate(AJ16, (360 * ° / a), T) → AJ17
  497. Rotate(AJ17, (360 * ° / a), T) → AJ18
  498. Rotate(AJ18, (360 * ° / a), T) → AJ19
  499. Rotate(AJ19, (360 * ° / a), T) → AJ20
  500. CircleSector(T, C14, AJ15) → AK15
  501. CircleSector(T, AJ15, AJ16) → AK16
  502. CircleSector(T, AJ16, AJ17) → AK17
  503. CircleSector(T, AJ17, AJ18) → AK18
  504. CircleSector(T, AJ18, AJ19) → AK19
  505. CircleSector(T, AJ19, AJ20) → AK20
  506. Rotate(k_1, (-(α - (29 * β))), C15) → q_2
  507. Center(q_2) → U
  508. Rotate(C15, (360 * ° / a), U) → AL16
  509. Rotate(AL16, (360 * ° / a), U) → AL17
  510. Rotate(AL17, (360 * ° / a), U) → AL18
  511. Rotate(AL18, (360 * ° / a), U) → AL19
  512. Rotate(AL19, (360 * ° / a), U) → AL20
  513. CircleSector(U, C15, AL16) → AM16
  514. CircleSector(U, AL16, AL17) → AM17
  515. CircleSector(U, AL17, AL18) → AM18
  516. CircleSector(U, AL18, AL19) → AM19
  517. CircleSector(U, AL19, AL20) → AM20
  518. Rotate(p_1, (-(α - (31 * β))), C16) → r_2
  519. Center(r_2) → V
  520. Rotate(C16, (360 * ° / a), V) → AN17
  521. Rotate(AN17, (360 * ° / a), V) → AN18
  522. Rotate(AN18, (360 * ° / a), V) → AN19
  523. Rotate(AN19, (360 * ° / a), V) → AN20
  524. CircleSector(V, C16, AN17) → AO17
  525. CircleSector(V, AN17, AN18) → AO18
  526. CircleSector(V, AN18, AN19) → AO19
  527. CircleSector(V, AN19, AN20) → AO20
  528. Rotate(q_1, (-(α - (33 * β))), C17) → s_2
  529. Center(s_2) → W
  530. Rotate(C17, (360 * ° / a), W) → AP18
  531. Rotate(AP18, (360 * ° / a), W) → AP19
  532. Rotate(AP19, (360 * ° / a), W) → AP20
  533. CircleSector(W, C17, AP18) → AQ18
  534. CircleSector(W, AP18, AP19) → AQ19
  535. CircleSector(W, AP19, AP20) → AQ20
  536. Rotate(r_1, (-(α - (35 * β))), C18) → t_2
  537. Center(t_2) → Z
  538. Rotate(C18, (360 * ° / a), Z) → AR19
  539. Rotate(AR19, (360 * ° / a), Z) → AR20
  540. CircleSector(Z, C18, AR19) → AS19
  541. CircleSector(Z, AR19, AR20) → AS20
  542. Rotate(s_1, (-(α - (37 * β))), C19) → c_3
  543. Center(c_3) → A_1
  544. Rotate(C19, (360 * ° / a), A_1) → AT20
  545. CircleSector(A_1, C19, AT20) → AU20
  546. CircleSector(C1, G2, B') → AV1
  547. CircleSector(C2, G3, G2) → AV2
  548. CircleSector(C3, G4, G3) → AV3
  549. CircleSector(C4, G5, G4) → AV4
  550. CircleSector(C5, G6, G5) → AV5
  551. CircleSector(C6, G7, G6) → AV6
  552. CircleSector(C7, G8, G7) → AV7
  553. CircleSector(C8, G9, G8) → AV8
  554. CircleSector(C9, G10, G9) → AV9
  555. CircleSector(C10, G11, G10) → AV10
  556. CircleSector(C11, G12, G11) → AV11
  557. CircleSector(C12, G13, G12) → AV12
  558. CircleSector(C13, G14, G13) → AV13
  559. CircleSector(C14, G15, G14) → AV14
  560. CircleSector(C15, G16, G15) → AV15
  561. CircleSector(C16, G17, G16) → AV16
  562. CircleSector(C17, G18, G17) → AV17
  563. CircleSector(C18, G19, G18) → AV18
  564. CircleSector(C19, G20, G19) → AV19
  565. Midpoint(C20, C21) → F21
  566. CircleSector(C20, G21, G20) → AV20
  567. Midpoint(A1, C1) → B_1
  568. Ray(B', B_1) → i
  569. Intersect(f, i, 1) → C_1
  570. Segment(B', C_1) → j
  571. Midpoint(D2, A1) → D_1
  572. Ray(F, D_1) → l
  573. Intersect(d', E1, 1) → E_1
  574. Intersect(d', E1, 1) → F_1
  575. Intersect(E1, l) → G_1,
  576. Segment(F, G_1) → m

AI Drawing Prompt Reference

Draw a circle with a center and mark the axes, finding their intersection. Mark points on the circle, then rotate these points around the center at equal angles to complete a full rotation. Move parts of the circle to the other side to form sectors. Connect the edges of adjacent sectors to create triangles and trapezoids. Use a slider to adjust the angle and demonstrate the derivation of the circle area formula.
The prompt above was compiled from the original drawing commands and is for reference only.