Innen

Außen

Wie\space groß\space ist\space eigentlich\space π?

Gegeben\space sei\space ein\space Kreis\space mit\space dem\space Radius\space r.\
Dieser\space ist\space mit\space dem\space Schieberegeler\space rechte\space zwischen\space 1\space LE\space und\space 10\space LE\space variabel.

Wir\space wissen,\space dass\space man\space den\space Flächeninhalt\space eines\space Kreises\ mit\space A=πr^2\space bestimmen\space kann.

Im\space Folgenden\space werden\space wir\space den\space Flächeninhalt\space des\space Kreises\ mit\space der\space Summe\space der\space Flächeninhalte\space mehrerer\space gelcihschenkliger\ Dreiecke\space ersetzen.

Die\space roten,\space inneren\space Dreiecke\space haben\space eine\space Schenkellänge,\ die\space genau\space dem\space Radius\space entspricht.\space Bei\space den\space dlauen,\space äußeren\ Dreiecken\space entspricht\space der\space Radius\space genau\space der Höhe\space im\space Scheitelpunkt.

Benutze\space die\space Bedinfelter\space 'Innen'\space und \space 'Außen'\space um\space dir\space wahlweise\ die\space roten\space oder\space die\space blauen\space Dreiecke\space anzeigen\space zu\space lassen,\space sowie\space den\ Schiberegler\space n\space um\space zu\space entscheiden,\space in\space wie\space viele\space Dreiecke\space der\space Kreis\ zerlegt\space werden\space soll.

Die\space rot\space oder\space blau\space markierte\space Fläche\space kann\space durch\space n\cdot A_\Delta \ berechnet\space werden.

Die\space Fläche\space eines\space Kreises\space wird\space durch\space π\cdot r^2\space berechnet.

Da\space beides\space übereinstimmen\space soll\space muss\ die\space folgende\space Aussage\space gelten:

n\cdot A_\Delta ≟ \pi\space\cdot r^2

⇔ \pi ≟ {n\cdot A_\Delta \over r^2}

Folglich\space\:ist\space einfach\space berechenbar,\space dass

n = 3 (slider from 2 to 15, step 1)
A = (0, 0)
Radius = 10 (slider from 1 to 10, step 0.1)
c = Circle(A, Radius)
B = Intersect(c, xAxis, 2)
B' = Rotate(B, (360 * ° / n), A)
α = Angle(B, A, B')
B'' = Rotate(B', α, A)
B''' = Rotate(B'', (360 * ° / n), A)
C = Rotate(B''', (360 * ° / n), A)
C' = Rotate(C, (360 * ° / n), A)
C'' = Rotate(C', (360 * ° / n), A)
C''' = Rotate(C'', (360 * ° / n), A)
D = Rotate(C''', (360 * ° / n), A)
D' = Rotate(D, (360 * ° / n), A)
D'' = Rotate(D', (360 * ° / n), A)
D''' = Rotate(D'', (360 * ° / n), A)
E = Rotate(D''', (360 * ° / n), A)
E' = Rotate(E, (360 * ° / n), A)
Z_2 = Midpoint(E', B)
A_1 = Midpoint(D'', D''')
f12 = Line(A, A_1)
T_2 = Intersect(c, f12, 2)
S_2 = Midpoint(E, B)
Q_2 = Midpoint(D''', B)
W = Midpoint(D', D)
f10 = Line(A, W)
N_2 = Intersect(c, f10, 2)
a = true
b = false
β = Angle(B', A, B'')
γ = Angle(B'', A, B''')
δ = Angle(B''', A, C)
ε = Angle(C, A, C')
ζ = Angle(C', A, C'')
η = Angle(C'', A, C''')
θ = Angle(C''', A, D)
ι = Angle(D, A, D')
κ = Angle(D', A, D'')
λ = Angle(D'', A, D''')
μ = Angle(D''', A, E)
ν = Angle(E, A, E')
E'' = Rotate(E', (360 * ° / n), A)
ξ = Angle(E', A, E'')
E''' = Rotate(E'', (360 * ° / n), A)
ο = Angle(E'', A, E''')
F = Rotate(E''', (360 * ° / n), A)
ρ = Angle(E''', A, F)
F' = Rotate(F, (360 * ° / n), A)
σ = Angle(F, A, F')
F'' = Rotate(F', (360 * ° / n), A)
τ = Angle(F', A, F'')
F''' = Rotate(F'', (360 * ° / n), A)
υ = Angle(F'', A, F''')
G = Rotate(F''', (360 * ° / n), A)
ϕ = Angle(F''', A, G)
G' = Rotate(G, (360 * ° / n), A)
χ = Angle(G, A, G')
G'' = Rotate(G', (360 * ° / n), A)
ψ = Angle(G', A, G'')
G''' = Rotate(G'', (360 * ° / n), A)
ω = Angle(G'', A, G''')
H = Rotate(G''', (360 * ° / n), A)
α_1 = Angle(G''', A, H)
H' = Rotate(H, (360 * ° / n), A)
β_1 = Angle(H, A, H')
H'' = Rotate(H', (360 * ° / n), A)
γ_1 = Angle(H', A, H'')
H''' = Rotate(H'', (360 * ° / n), A)
δ_1 = Angle(H'', A, H''')
I = Rotate(H''', (360 * ° / n), A)
ε_1 = Angle(H''', A, I)
I' = Rotate(I, (360 * ° / n), A)
ζ_1 = Angle(I, A, I')
d02 = Line(B', A)
d03 = Line(B'', A)
V1_1, V1a, b_1, V1b = Polygon(B, B', A)
V2, V2a, b'_1, V2b = Polygon(B', B'', A)
V2_x, V2_xa, V3b, b_2 = Polygon(B'', B, A)
O = Midpoint(B', B)
P = Midpoint(B', B'')
Q = Midpoint(B'', B''')
R = Midpoint(B''', C)
S = Midpoint(C, C')
T = Midpoint(C'', C')
U = Midpoint(C''', C'')
V_1 = Midpoint(C''', D)
Z = Midpoint(D'', D')
B_1 = Midpoint(E, D''')
C_1 = Midpoint(E, E')
D_1 = Midpoint(E'', E')
E_1 = Midpoint(E''', E'')
F_1 = Midpoint(F, E''')
G_1 = Midpoint(F', F)
H_1 = Midpoint(F'', F')
I_1 = Midpoint(F''', F'')
J_1 = Midpoint(G, F''')
K_1 = Midpoint(G', G)
L_1 = Midpoint(G'', G')
M_1 = Midpoint(G''', G'')
N_1 = Midpoint(H, G''')
O_1 = Midpoint(H', H)
P_1 = Midpoint(H'', H')
Q_1 = Midpoint(H''', H'')
R_1 = Midpoint(I, H''')
S_1 = Midpoint(I', I)
R_3 = Midpoint(B, I')
f02 = OrthogonalLine(O, V1a)
N_4 = Intersect(c, f02, 2)
g01 = Line(N_4, V1a)
J = Intersect(d02, g01)
K = Intersect(g01, xAxis)
W1, k, W1a, j = Polygon(A, J, K)
f03 = OrthogonalLine(P, V2a)
L = Intersect(c, f03, 1)
g02 = Line(L, V2a)
M = Intersect(g02, d03)
W2, j_1, m, W2a = Polygon(M, A, J)
N = Midpoint(B'', B)
T_1 = Intersect(c, d02, 2)
W2_x, W2_xa, m_1, k_1 = Polygon(M, K, A)
d04 = Line(A, B''')
d05 = Line(A, C)
d06 = Line(A, C')
d07 = Line(A, C'')
d08 = Line(A, C''')
d09 = Line(A, D)
d10 = Line(A, D')
d11 = Line(A, D'')
d12 = Line(A, D''')
d13 = Line(A, E)
d14 = Line(A, E')
d15 = Line(A, E'')
d16 = Line(A, E''')
d17 = Line(A, F)
d18 = Line(A, F')
d19 = Line(A, F'')
d20 = Line(A, F''')
d21 = Line(A, G)
d22 = Line(A, G')
d23 = Line(A, G'')
d24 = Line(A, G''')
d25 = Line(A, H)
d26 = Line(A, H')
d27 = Line(A, H'')
d28 = Line(A, H''')
d29 = Line(A, I)
d30 = Line(A, I')
f04 = Line(A, Q)
f05 = Line(A, R)
f06 = Line(A, S)
f07 = Line(A, T)
f08 = Line(A, U)
f09 = Line(A, V_1)
f11 = Line(A, Z)
f13 = Line(A, B_1)
f14 = Line(A, C_1)
f15 = Line(A, D_1)
f16 = Line(A, E_1)
f17 = Line(A, F_1)
f18 = Line(A, G_1)
f19 = Line(A, H_1)
f20 = Line(A, I_1)
f21 = Line(A, J_1)
f22 = Line(A, K_1)
f23 = Line(A, L_1)
f24 = Line(A, M_1)
f25 = Line(A, N_1)
f26 = Line(A, O_1)
f27 = Line(A, P_1)
f28 = Line(A, Q_1)
f29 = Line(A, R_1)
f30 = Line(A, S_1)
f30end = Line(A, R_3)
V3, V3a, b''_2, V2_xb = Polygon(B'', B''', A)
V3_x, a_1, b''', b_3 = Polygon(B''', B, A)
V = Intersect(c, f04, 1)
g03 = OrthogonalLine(V, f04)
U_1 = Intersect(g03, d04)
W3, m_2, u_1, a_2 = Polygon(U_1, A, M)
W3_x, a_3, u_2, k_2 = Polygon(U_1, K, A)
V4, a_4, b'''_1, c_1 = Polygon(B''', C, A)
V4_x, a_5, c_2, b_4 = Polygon(C, B, A)
W_1, W_2 = Intersect(c, f05)
g04 = OrthogonalLine(W_2, f05)
Z_1 = Intersect(g04, d05)
W4, u_3, z_1, a_6 = Polygon(Z_1, A, U_1)
W4_x, a_7, z_2, k_3 = Polygon(Z_1, K, A)
V5, a_8, c_3, c' = Polygon(C, C', A)
V5_x, a_9, c'_1, b_5 = Polygon(C', B, A)
A_2, A_3 = Intersect(c, f06)
B_2 = Intersect(f03, f06)
g05 = OrthogonalLine(A_3, f06)
C_2 = Intersect(g05, d06)
W5, a_{10}, z_3, c_4 = Polygon(Z_1, C_2, A)
W5_x, a_{11}, c_5, k_4 = Polygon(C_2, K, A)
D_2 = Midpoint(C'', B)
e1 = Line(A, D_2)
V6, a_{12}, c'_2, c'' = Polygon(C', C'', A)
V6_x, a_{13}, c''_1, b_6 = Polygon(C'', B, A)
E_2 = Intersect(c, f07, 2)
g06 = OrthogonalLine(E_2, f07)
F_2 = Intersect(d07, g06)
W6, a_{14}, c_6, f_2 = Polygon(C_2, F_2, A)
W6_x, a_{15}, f_1, k_5 = Polygon(F_2, K, A)
e2 = Line(A, W_1)
V7, a_{16}, c''_2, c''' = Polygon(C'', C''', A)
V7_x, a_{17}, c'''_1, b_7 = Polygon(C''', B, A)
G_2 = Intersect(c, f08, 2)
g07 = OrthogonalLine(G_2, f08)
H_2 = Intersect(d08, g07)
W7, a_{18}, f_3, h_2 = Polygon(F_2, H_2, A)
W7_x, a_{19}, h_1, k_6 = Polygon(H_2, K, A)
I_2 = Midpoint(D, B)
e3 = Line(A, I_2)
V8, a_{20}, c'''_2, d = Polygon(C''', D, A)
V8_x, a_{21}, d_1, b_8 = Polygon(D, B, A)
J_2 = Intersect(c, f09, 2)
g08 = OrthogonalLine(J_2, f09)
K_2 = Intersect(d09, g08)
W8, a_{22}, h_3, k_7 = Polygon(H_2, K_2, A)
W8_x, a_{23}, k_8, k_9 = Polygon(K_2, K, A)
V9, a_{24}, d_2, d' = Polygon(D, D', A)
V9_x, a_{25}, d'_1, b_9 = Polygon(D', B, A)
g09 = OrthogonalLine(N_2, f10)
L_2 = Intersect(d10, g09)
W9, a_{26}, k_{10}, l_2 = Polygon(K_2, L_2, A)
W9_x, a_{27}, l_1, k_{11} = Polygon(L_2, K, A)
M_2 = Midpoint(D'', B)
V10, a_{28}, d'_2, d'' = Polygon(D', D'', A)
V10_x, a_{29}, d''_1, b_{10} = Polygon(D'', B, A)
O_2 = Intersect(c, f11, 2)
g10 = OrthogonalLine(O_2, f11)
P_2 = Intersect(d11, g10)
W10, a_{30}, l_3, p_2 = Polygon(L_2, P_2, A)
W10_x, a_{31}, p_1, k_{12} = Polygon(P_2, K, A)
V11, a_{32}, d''_2, d''' = Polygon(D'', D''', A)
V11_x, a_{33}, d'''_1, b_{11} = Polygon(D''', B, A)
g11 = OrthogonalLine(T_2, f12)
R_2 = Intersect(d12, g11)
W11, a_{34}, p_3, r_2 = Polygon(P_2, R_2, A)
W11_x, a_{35}, r_1, k_{13} = Polygon(R_2, K, A)
U_2 = Intersect(c, f13, 2)
V12, a_{36}, d'''_2, e = Polygon(D''', E, A)
V12_x, a_{37}, e_1, b_{12} = Polygon(E, B, A)
g12 = OrthogonalLine(U_2, f13)
V_2 = Intersect(d13, g12)
W12, a_{38}, r_3, v_2 = Polygon(R_2, V_2, A)
W12_x, a_{39}, v_1, k_{14} = Polygon(V_2, K, A)
B_3 = Intersect(c, f14, 2)
V13, a_{40}, e_2, e' = Polygon(E, E', A)
V13_x, a_{41}, e'_1, b_{13} = Polygon(E', B, A)
g13 = OrthogonalLine(B_3, f14)
C_3 = Intersect(d14, g13)
W13, a_{42}, v_3, c_7 = Polygon(V_2, C_3, A)
W13_x, a_{43}, c_8, k_{15} = Polygon(C_3, K, A)
D_3 = Midpoint(B, E'')
E_3 = Intersect(c, f15, 2)
e4 = Line(A, D_3)
V14, a_{44}, e'_2, e'' = Polygon(E', E'', A)
V14_x, a_{45}, e''_1, b_{14} = Polygon(E'', B, A)
g14 = OrthogonalLine(E_3, f15)
F_3 = Intersect(d15, g14)
W14, a_{46}, c_9, f_4 = Polygon(C_3, F_3, A)
W14_x, a_{47}, f_5, k_{16} = Polygon(F_3, K, A)
Text7: "Flächeninhalt\space der\space roten\space Fläche:\space " + (LaTeX[(n * V1_1)]) + "FE"
Text8: "Flächeninhalt\space des\space Kreises:\space" + (LaTeX[((π * Radius) * Radius)]) + "FE"
Text9: "Flächeninhalt\space der\space blauen\space Fläche:\space " + (LaTeX[(n * W1)]) + "FE "
V1 = NaN
Text16: "{" + (LaTeX[n]) + " \cdot " + (LaTeX[V1_1]) + " \over " + (LaTeX[Radius]) + "^2} = " + (LaTeX[(n * V1_1 / Radius / Radius)]) + " < \pi < " + (LaTeX[(n * W1 / Radius / Radius)]) + " ={" + (LaTeX[n]) + "\cdot " + (LaTeX[W1]) + " \over " + (LaTeX[Radius]) + "^2}"