求最小面积
面积为1的正n(n=3,4,5,...)边形, 将其放入直角三角形中, 求直角三角形最小面积。 数据A——面积为1的正n(n=3,4,5,...)边形, 将其放入直角三角形中, 且一条边位于直角边上, 三角形最小面积A(n)。数据B——面积为1的正n(n=3,4,5,...)边形, 将其放入直角等腰三角形中, 且一条边位于等腰三角形底边上, 三角形最小面积B(n)。
数据C——面积为1的正n(n=3,4,5,...)边形, 将其放入等腰三角形中, 且一条边位于等腰三角形底边上, 三角形最小面积C(n)。
数据A——{2.00000, 2.00000, 1.80902, 2.08333, 2.13203, 1.75888, 2.04329, 1.90623, 1.85385, 2.01036, 1.85105, 1.88833, 1.91008, 1.83134, 1.90182, 1.87171}
数据B——{1.73205, 2.25000, 1.80058, 1.91759, 1.94378, 1.75888, 1.90682, 1.87566, 1.84364, 1.88835, 1.84692, 1.86541, 1.87316, 1.83134, 1.86913, 1.86133}
数据C——{1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716}
数据B是这样来的——Table + Sqrt Sin[((2 # + 1) Pi)/n - Pi/4] & /@Range]^2*(2 Csc)/n], {n, 3, 18}]
数据D——面积为1的正n(n=3,4,5,...)边形, 将其放入三角形中, 三角形最小面积D(n)——天马行空——D(n) = C(n)。
谢谢各位！新年快乐！！！ 简化题目——面积为1的正n(n=3,4,5,...)边形, 将其放入三角形中,
三角形最小面积——{1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716}
问题1——来个反例。
问题2——提供资料。
问题3——通项公式。
谢谢各位！新年快乐！！！ 简化题目——面积为1的正n(n=3,4,5,...)边形, 将其放入三角形中,
三角形最小面积——{1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716, 1.66455, 1.66275, 1.64163, 1.66180, 1.66064, 1.64453, 1.66001, 1.65921}
这个没找到通项公式——但若把n=3,6,9,...拉出来,
{1.00000, 1.50000, 1.58626, 1.61603, 1.62973, 1.63716, 1.64163, 1.64453, 1.64652, 1.64794, 1.64899, 1.64979, 1.65041, 1.65090, 1.65130, 1.65162, 1.65189, 1.65212, 1.65231, 1.65247, 1.65262, 1.65274, 1.65284, 1.65294, 1.65302, 1.65309}
可以有个公式——Table Max)] (Abs] + (Cos - Cos[#])/Sqrt) & /@ ((2 Range - 1) Pi/(3 n))]^2, 6], {n, 30}]——虽然不满意。
慢慢长大,极限 = \(\frac{3\sqrt{3}}{\pi}\) = 1.6539866862653761485。 {1.00000, 1.50000, 1.58626, 1.61603, 1.62973, 1.63716, 1.64163, 1.64453, 1.64652, 1.64794, 1.64899, 1.64979, 1.65041, 1.65090, 1.65130, 1.65162, 1.65189, 1.65212, 1.65231, 1.65247, 1.65262, 1.65274, 1.65284, 1.65294, 1.65302, 1.65309}
这个可以有一个很漂亮的公式—— \(\D\cot\big(\frac{\pi}{3n}\big)\frac{\sqrt{3}}{n}\)
慢慢长大,极限 = \(\frac{3\sqrt{3}}{\pi}\) = 1.6539866862653761485。 简化题目——面积为1的正n(n=3,4,5,...)边形, 将其放入三角形中,
三角形最小面积——{1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716, 1.66455, 1.66275, 1.64163, 1.66180, 1.66064, 1.64453, 1.66001, 1.65921}
通项公式是这样——Table Cot/n, Mod == 0}, {8 Cos[(n + 6 - 4 Mod) Pi/(6 n)] (Cos + Sin[(n + 6 - 4 Mod) Pi/(6 n)])/(n Sin), Mod ≠ 0}}], {n, 3, 60}]
1.00000, 2.00000, 1.78885, 1.50000, 1.74224, 1.70711, 1.58626, 1.69443, 1.68239, 1.61603, 1.67720, 1.67167, 1.62973, 1.66905, 1.66606, 1.63716, 1.66455, 1.66275, 1.64163, 1.66180, 1.66064, 1.64453, 1.66001, 1.65921, 1.64652, 1.65876,
1.65819, 1.64794, 1.65787, 1.65745, 1.64899, 1.65721, 1.65689, 1.64979, 1.65670, 1.65645, 1.65041, 1.65630, 1.65611, 1.65090, 1.65599, 1.65583, 1.65130, 1.65573, 1.65560, 1.65162, 1.65552, 1.65542, 1.65189, 1.65535, 1.65526, 1.65212,
1.65520, 1.65513, 1.65231, 1.65508, 1.65502, 1.65247, 1.65498, 1.65492, 1.65262, 1.65488, 1.65484, 1.65274, 1.65481, 1.65476, 1.65284, 1.65474, 1.65470, 1.65294, 1.65468, 1.65464, 1.65302, 1.65462, 1.65459, 1.65309, 1.65457, 1.65455,
1.65316, 1.65453, 1.65451, 1.65322, 1.65449, 1.65447, 1.65327, 1.65446, 1.65444, 1.65331, 1.65443, 1.65441, 1.65336, 1.65440, 1.65439, 1.65340, 1.65438, 1.65436, 1.65343, 1.65435, 1.65434, 1.65346, 1.65433, 1.65432, 1.65349, 1.65431}
复杂的数据是这样:
{1,
2,
4/Sqrt,
3/2,
8/7 (Cos[\/7] + Sin[(3 \)/14]),
1 + 1/Sqrt,
Cot[\/9]/Sqrt,
2/5 (2 + Sqrt),
4/11 (Cos[\/22] + 2 Cos[(5 \)/22]) Csc[(2 \)/11],
1/4 (3 + 2 Sqrt),
4/13 Csc[(2 \)/13] (2 Cos[(3 \)/26] + Sin[(3 \)/13]),
4/7 Cot[\/7] (Cos[\/14] + Sin[\/7]),
1/5 Sqrt Cot[\/15],
1/8 (Sqrt + 4 Cos[\/8]) Csc[\/8],
4/17 (Cos[(3 \)/34] + 2 Cos[(7 \)/34]) Csc[(2 \)/17],
Cot[\/18]/(2 Sqrt),
4/19 (2 Cos[(5 \)/38] + Cos[(9 \)/38]) Csc[(2 \)/19],
1/5 (3 + Sqrt + Sqrt]),
1/7 Sqrt Cot[\/21],
2/11 (2 Cos[(3 \)/22] + Cos[(5 \)/22]) Csc[\/11],
4/23 (Cos[(5 \)/46] + 2 Cos[(9 \)/46]) Csc[(2 \)/23],
1/8 Sqrt Cot[\/24],
4/25 (2 Cos[(7 \)/50] + Cos[(11 \)/50]) Csc[(2 \)/25],
2/13 (Cos[(3 \)/26] + 2 Cos[(5 \)/26]) Csc[\/13],
Cot[\/27]/(3 Sqrt),
1/7 (2 Cos[\/7] + Cos[(3 \)/14]) Csc[\/14],
4/29 (Cos[(7 \)/58] + 2 Cos[(11 \)/58]) Csc[(2 \)/29],
1/10 Sqrt Cot[\/30],
4/31 (2 Cos[(9 \)/62] + Cos[(13 \)/62]) Csc[(2 \)/31],
1/8 (Cos[\/8] + 2 Cos[(3 \)/16]) Csc[\/16],
1/11 Sqrt Cot[\/33],
2/17 (2 Cos[(5 \)/34] + Cos[(7 \)/34]) Csc[\/17],
4/35 (Cos[(9 \)/70] + 2 Cos[(13 \)/70]) Csc[(2 \)/35],
Cot[\/36]/(4 Sqrt),
4/37 (2 Cos[(11 \)/74] + Cos[(15 \)/74]) Csc[(2 \)/37],
2/19 (Cos[(5 \)/38] + 2 Cos[(7 \)/38]) Csc[\/19],
1/13 Sqrt Cot[\/39],
1/40 (1 + Sqrt + 8 Cos[(3 \)/20]) Csc[\/20],
4/41 (Cos[(11 \)/82] + 2 Cos[(15 \)/82]) Csc[(2 \)/41],
1/14 Sqrt Cot[\/42],
4/43 (2 Cos[(13 \)/86] + Cos[(17 \)/86]) Csc[(2 \)/43],
1/11 (Cos[(3 \)/22] + 2 Cos[(2 \)/11]) Csc[\/22],
Cot[\/45]/(5 Sqrt),
2/23 (2 Cos[(7 \)/46] + Cos[(9 \)/46]) Csc[\/23],
4/47 (Cos[(13 \)/94] + 2 Cos[(17 \)/94]) Csc[(2 \)/47],
1/16 Sqrt Cot[\/48],
4/49 (2 Cos[(15 \)/98] + Cos[(19 \)/98]) Csc[(2 \)/49],
2/25 (Cos[(7 \)/50] + 2 Cos[(9 \)/50]) Csc[\/25],
1/17 Sqrt Cot[\/51],
1/13 (2 Cos[(2 \)/13] + Cos[(5 \)/26]) Csc[\/26],
4/53 (Cos[(15 \)/106] + 2 Cos[(19 \)/106]) Csc[(2 \)/53],
Cot[\/54]/(6 Sqrt),
4/55 (2 Cos[(17 \)/110] + Cos[(21 \)/110]) Csc[(2 \)/55],
1/14 (Cos[\/7] + 2 Cos[(5 \)/28]) Csc[\/28],
1/19 Sqrt Cot[\/57],
2/29 (2 Cos[(9 \)/58] + Cos[(11 \)/58]) Csc[\/29],
4/59 (Cos[(17 \)/118] + 2 Cos[(21 \)/118]) Csc[(2 \)/59],
1/20 Sqrt Cot[\/60]}
问题1——来个反例。
问题2——提供资料。
问题3——通项公式。
谢谢各位！新年快乐！！！ 同理——边长为1的正n(n=3,4,5,...)边形, 将其放入三角形中,
三角形最小面积——0.433013, 2.00000, 3.07768, 3.89711, 6.33114, 8.24264, 9.80596, 13.0373, 15.7567, 18.0933, 22.1151, 25.6343, 28.7523, 33.5635, 37.8786, 41.7815, 47.3817, 52.4910, 57.1804, 63.5696, 69.4717, 74.9486, 82.1269, 88.8212}
取整数——{0, 2, 3, 3, 6, 8, 9, 13, 15, 18, 22, 25, 28, 33, 37, 41, 47, 52, 57, 63, 69, 74, 82, 88, 95, 103, 110, 117, 126, 134, 142, 152, 161, 169, 180, 189, 199, 210, 221, 231, 243, 254, 265, 278, 290, 302, 315, 328, 341, 355, 369, 382, 398, 412}
通项公式是这样——Table/4) Cot^2,3n}, {Cos[(n + 6 - 4 Mod) Pi/(6 n)] (Cos + Sin[(n + 6 - 4 Mod) Pi/(6 n)])/Sin^2, True}}], 6], {n, 3, 60}] 同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正方形形中,
三角形最小面积——{0.933013, 1.00000, 2.55397, 3.73205, 4.98562, 5.82843, 8.22788, 10.2159, 12.2807, 13.9282, 17.1441, 19.9425, 22.8181, 25.2741, 29.3027, 32.9115, 36.5978, 39.8635, 44.7035, 49.1229, 53.6197, 57.6955, 63.3466, 68.5765}
取整数——{0, 1, 2, 3, 4, 5, 8, 10, 12, 13, 17, 19, 22, 25, 29, 32, 36, 39, 44, 49, 53, 57, 63, 68, 73, 78, 85, 91, 97, 103, 110, 117, 124, 130, 138, 146, 154, 161, 170, 178, 187, 195, 205, 214, 223, 232, 243, 253, 263, 273, 284, 295, 306, 317, 329}
通项公式是这样——Table, (Cos^2 Cos^2)/(Cos Sin), 4\n, 1/Tan, True, (1 + Cos)/Sin] 1/Tan], {n, 3, 60}]
补充内容 (2026-2-24 11:38):
正方形最小面积——{0.933013, 1.00000, 2.55397, 3.73205, 同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正五边形形中, 三角形最小面积——
补充内容 (2026-2-23 09:38):
同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正五边形中, 正五边形最小面积 同理——边长为1的正n(n=3,4,5,...)边形, 将其放入正五边形中, 正五边形最小面积——
{1.05047, 1.51007, 1.72048, 3.36936, 4.70078, 6.11731, 7.50981, 8.60239, 11.1712, 13.4755, 15.7387, 18.0917, 20.1012, 23.5935, 26.7802, 30.0007, 33.2554, 37.1113, 40.6168,
44.7232, 48.864, 53.0391, 56.9066, 62.2601, 67.2674, 72.3647, 77.4238, 82.2111, 88.4659, 94.4489, 100.394, 106.429, 112.117, 119.292, 126.158, 133.06, 139.996, 147.531, 154.719,
162.505, 170.327, 178.183, 185.731, 194.766, 203.453, 212.231, 220.971, 229.439, 239.375, 249.038, 258.664, 268.38, 277.748, 288.604, 299.152, 309.734, 320.35, 331.567}
Table/2; v = Csc/2*{Cos[#], Sin[#]} & /@Table; m = Table]; v2 = v[ + 1]]; d = v2 - v1;
{-d[], d[]}, {i, 5}]; w=Flatten@Table[(x+ R Cos[\+ 2 Pi k/n] - a v[]) m[]+ (y + R Sin[\+ 2 Pi k/n] - a v[]) m[] ≤ 0, {k, 0, n - 1}, {i, 5}];
w = Join; s = FindMinimum[{a, w}, {{a, 3 R}, {x, 0}, {y, 0}, {\, 0}}, Method -> "InteriorPoint", MaxIterations -> 1000]; 5 Cot (a /. s[])^2/4], {n, 3, 30}]
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