圆面积蒙特卡洛模拟求π

几何画板及绘图内容加载中，请稍等……

2023-03-13 23:14

所属专题 Pi Approximations

①本范例通过蒙特卡洛方法模拟随机点落在圆内，演示圆面积与π的计算；②涉及核心数学概念：概率统计、几何面积、极限思想；③作图要点：使用Polygon, Circle, Sequence, Take, Intersect, Segment, Midpoint, CountIf等命令绘制随机点、圆、计算交点并计数；④适合高中数学教学，用于理解概率与几何结合的π值估算方法。

GeoGebra 命令及参数

Polygon(A, B, 4) → poly1, f, g, h, i, C, D
Circle(E, 1) → c
Sequence(RandomPointIn[poly1], i, 1, 20000) → l1, {(1.4308454114691973, 0.015245113107956065), (1.4973399403383731, 1.6382036538725882), (1.0070817542709698, 1.1532354449664712), (0.19137016500287474, 1.8581750793070622), (0.777671544478433, 0.018303904985739194), (0.2632161607143204, 0.23287378617607124), (1.9002346214796269, 1.0075875820437272), (1.7427367618390972, 1.0260859042534058), (1.7333456774927043, 0.8556859285897704), (1.2587190134170427, 1.125481271219617), (0.7348486270062582, 1.017898194661225), (0.7416063258713428, 1.1886157004046378), (1.9331254743616135, 0.9763448121266434), (1.6120534933493669, 0.5165355666783216), (0.4577083420860766, 1.696084576838331), (0.7082984887667014, 1.8752597527518762), (0.16445198137948336, 0.08529278522604944), (0.0354207198848302, 1.6436189298316688), (1.001812742155773, 0.503017692683202), (0.312132204083696, 0.38663378974533313), (0.43860660668881524, 0.10998758858438129), (0.24582666326540448, 1.2368172965357096), (1.0099489122299974, 1.17239181081306), (1.4841903346261194, 0.31148976417241436), (1.8369047110014154, 1.3055133932348435), (1.2717551622902183, 0.8023982523592614), (0.8367081952570528, 0.0835445504975969), (0.9823544791647345, 0.344634328290361), (0.26423730345957064, 1.4438676041773795), (0.903517943424937, 1.805048784593365), (1.21175698135604, 1.8785393222922275), (1.916422737480621, 0.7260635450141539), (0.7427743406658189, 0.5326673379009326), (1.0450933754353948, 0.3189849450496889), (0.4471751777533033, 1.018823196459785), (0.24112326914614957, 1.1216258048878884), (1.3786412568141457, 0.00858677797909313), (0.2252675488436484, 1.2182460378513127), (1.571674333802998, 0.47677243526712143), (1.7248354956311758, 0.26552558166950146), (0.6343819696314161, 1.850838825742626), (1.025889190792629, 0.4764860730791911), (1.2745082671945296, 1.2574380817052742), (1.4018322034817887, 0.18498051779770086), (0.040020153974641204, 1.560013010169146), (0.6850516916931755, 0.6465745789370851), 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Take(l1, 1, n) → l2
Intersect(c, i, 1) → F
Segment(F, E) → j
Midpoint(j) → G
CountIf(Distance[E, X] ≤ 1, X, l2) → a