Circumference (Circonférence)

RADIUS (rayon)

lancer animation !

Reset

Pause

RADIUS (rayon)

AIRE D'UN   CERCLE  

Questions:

1) Comment l'aire du TRIANGLE (à droite) se compare-t-elle à l'aire du CERCLE (à gauche) ?

2) Utilisez ce que vous venez d'observer pour écrire une formule pour l'AIRE D'UN CERCLE en fonction de sa CIRCONFÉRENCE et de son RAYON.                    

3) Réécrivez votre formule en (2) afin que la SURFACE D'UN CERCLE soit écrite uniquement en fonction de son RAYON.  

a = 26 (slider from 0 to 26, step 0.01)
e_2 = If(2 < a < 3, Translate[h, (((0.9 * π) * (a - 2)), 0)])
Q_1 = Center(e_2)
A = (-3.141592653589793, 0)
B = (3.141592653589793, 0)
a_1 = Segment(A, B)
D = (0, 0)
D' = Dilate(D, If[0 < a < 1, 1 - a, 1], B)
C: D' + (0, 1)
E = Intersect(c, yAxis, 2)
F = Point(yAxis)
c' = Translate(c, If[0 < a < 1, ((π * a), 0)])
G = Center(c')
b: x = x(G)
d = Segment(D, D')
D'_1 = Mirror(D, f)
D'' = Mirror(D', f)
d' = Segment(D'_1, D'')
J = Intersect(g, f, 2)
K = (2.9845130209103035, 0.05)
L = (2.827433388230814, 0.1)
N = (2.67035, 0.15)
P = (2.5132741228718345, 0.2)
Q = (2.356194490192345, 0.25)
R = (2.199114857512855, 0.3)
S = (2.0420352248333655, 0.35)
T = (1.8849555921538759, 0.4)
U = (1.7278759594743864, 0.45)
V = (1.5707963267948966, 0.5)
W = (1.413716694115407, 0.55)
Z = (1.2566370614359172, 0.6)
A_1 = (1.0995574287564276, 0.65)
B_1 = (0.9424777960769379, 0.7)
C_1 = (0.7853981633974483, 0.75)
D_1 = (0.6283185307179586, 0.8)
E_1 = (0.47123889803846897, 0.85)
F_1 = (0.3141592653589793, 0.9)
G_1 = (0.15707963267948966, 0.95)
K' = Mirror(K, f)
L' = Mirror(L, f)
N' = Mirror(N, f)
P' = Mirror(P, f)
Q' = Mirror(Q, f)
R' = Mirror(R, f)
S' = Mirror(S, f)
T' = Mirror(T, f)
U' = Mirror(U, f)
V' = Mirror(V, f)
W' = Mirror(W, f)
Z' = Mirror(Z, f)
M = Mirror(A_1, f)
O = Mirror(B_1, f)
H_1 = Mirror(C_1, f)
I_1 = Mirror(D_1, f)
J_1 = Mirror(E_1, f)
K_1 = Mirror(F_1, f)
L_1 = Mirror(G_1, f)
J' = Dilate(J, If[1 < a < 2, 2 - a, 1], K)
t_1 = If(1 < a < 2, Translate[g, (((0.95 * π) * (a - 1)), 0)])
N_1 = Center(t_1)
l: x = x(N_1)
j = Segment(J, J')
J''_1 = Mirror(J', f)
b_1 = Segment(J, J''_1)
J'_1 = Mirror(J, f)
J''_2 = Mirror(J', f)
m = Segment(K', K)
i_1: x = x(Q_1)
I = Intersect(g, f, 1)
M_1, O_1 = Intersect(h, f)
P_1, R_1 = Intersect(k, f)
S_1, T_1 = Intersect(p, f)
U_1 = If(1 < a < 2, Translate[I, (((0.95 * π) * (a - 1)), 0)])
V_1 = Rotate(U_1, ((1 - a) * 180°), N_1)
c_2 = CircleArc(N_1, J', V_1)
d_2 = Mirror(c_2, f)
W_1 = If(2 < a < 3, Translate[M_1, (((0.9 * π) * (a - 2)), 0)])
Z_1 = Dilate(O_1, If[2 < a < 3, 3 - a, 1], L)
A_2 = Rotate(W_1, ((2 - a) * 180°), Q_1)
f_2 = CircleArc(Q_1, Z_1, A_2)
n = Segment(O_1, Z_1)
g_2 = Mirror(f_2, f)
B_2 = Mirror(O_1, f)
C_2 = Mirror(Z_1, f)
n' = Segment(B_2, C_2)
j_1 = Segment(L', L)
l_1 = Segment(N', N)
D_2 = Intersect(k, f, 1)
E_2 = Dilate(R_1, If[3 < a < 4, 4 - a, 1], N)
h_2 = If(3 < a < 4, Translate[k, (((0.85 * π) * (a - 3)), 0)])
F_2 = Center(h_2)
G_2 = If(3 < a < 4, Translate[D_2, (((0.85 * π) * (a - 3)), 0)])
H_2 = Rotate(G_2, ((3 - a) * 180°), F_2)
k_2 = CircleArc(F_2, E_2, H_2)
m_1 = Segment(R_1, E_2)
p_2 = Mirror(k_2, f)
I_2 = Mirror(R_1, f)
J_2 = Mirror(E_2, f)
n_1 = Segment(I_2, J_2)
K_2 = Intersect(f, yAxis)
L_2 = Intersect(f, yAxis)
M_2 = Intersect(f, yAxis)
N_2, O_2 = Intersect(c, f)
P_2 = If(0 < a < 1, Translate[N_2, ((a * π), 0)])
H = Rotate(P_2, If[0 < a < 1, ((-a) * 180°)], C)
e = CircleArc(C, D', H)
e' = Mirror(e, f)
Q_2, R_2 = Intersect(q, f)
S_2 = Intersect(f, yAxis)
T_2 = Intersect(f, yAxis)
U_2 = Intersect(f, yAxis)
V_2 = Intersect(f, yAxis)
W_2 = Intersect(f, yAxis)
Z_2 = Intersect(f, yAxis)
A_3 = Intersect(f, yAxis)
B_3 = Intersect(r, s)
C_3 = Intersect(f, yAxis)
D_3 = Intersect(r, f, 1)
E_3 = Intersect(s, f, 1)
F_3 = Intersect(t, f, 1)
G_3 = Intersect(c_1, f, 1)
H_3 = Intersect(d_1, f, 1)
I_3 = Intersect(e_1, f, 1)
J_3 = Intersect(f_1, f, 1)
K_3 = Intersect(g_1, f, 1)
L_3 = Intersect(h_1, f, 1)
M_3 = Intersect(k_1, f, 1)
N_3 = Intersect(p_1, f, 1)
O_3 = Intersect(q_1, f, 1)
P_3 = Intersect(r_1, f, 1)
Q_3 = Intersect(s_1, f, 1)
R_3 = Intersect(s_1, f, 2)
S_3 = Intersect(r_1, f, 2)
T_3 = Intersect(q_1, f, 2)
U_3 = Intersect(f, yAxis)
V_3 = Intersect(p_1, f, 2)
W_3 = Intersect(k_1, f, 2)
Z_3 = Intersect(h_1, f, 2)
A_4 = Intersect(g_1, f, 2)
B_4 = Intersect(f_1, f, 2)
C_4 = Intersect(e_1, f, 2)
D_4 = Intersect(d_1, f, 2)
E_4 = Intersect(c_1, f, 2)
F_4 = Intersect(t, f, 2)
G_4 = Intersect(s, f, 2)
H_4 = Intersect(r, f, 2)
I_4 = Dilate(T_1, If[4 < a < 5, 5 - a, 1], P)
J_4 = Dilate(R_2, If[5 < a < 6, 6 - a, 1], Q)
a_2 = Segment(T_1, I_4)
K_4 = Mirror(T_1, f)
L_4 = Mirror(I_4, f)
b_2 = Segment(K_4, L_4)
i_2 = Segment(P', P)
j_2 = Segment(Q', Q)
l_2 = Segment(R', R)
m_2 = Segment(S', S)
n_2 = Segment(T', T)
q_2 = Segment(U', U)
r_2 = Segment(V', V)
s_2 = Segment(W', W)
t_2 = Segment(Z', Z)
a_3 = Segment(M, A_1)
b_3 = Segment(O, B_1)
c_3 = Segment(H_1, C_1)
d_3 = Segment(I_1, D_1)
e_3 = Segment(J_1, E_1)
f_3 = Segment(K_1, F_1)
g_3 = Segment(L_1, G_1)
h_3 = Segment(R_2, J_4)
M_4 = Dilate(H_4, If[6 < a < 7, 7 - a, 1], R)
i_3 = Segment(H_4, M_4)
N_4 = Dilate(G_4, If[7 < a < 8, 8 - a, 1], S)
j_3 = Segment(G_4, N_4)
O_4 = Dilate(F_4, If[8 < a < 9, 9 - a, 1], T)
k_3 = Segment(F_4, O_4)
P_4 = Dilate(E_4, If[9 < a < 10, 10 - a, 1], U)
l_3 = Segment(E_4, P_4)
Q_4 = Dilate(D_4, If[10 < a < 11, 11 - a, 1], V)
m_3 = Segment(D_4, Q_4)
R_4 = Dilate(C_4, If[11 < a < 12, 12 - a, 1], W)
n_3 = Segment(C_4, R_4)
S_4 = Dilate(B_4, If[12 < a < 13, 13 - a, 1], Z)
p_3 = Segment(B_4, S_4)
T_4 = Dilate(A_4, If[13 < a < 14, 14 - a, 1], A_1)
q_3 = Segment(A_4, T_4)
U_4 = Dilate(Z_3, If[14 < a < 15, 15 - a, 1], B_1)
r_3 = Segment(Z_3, U_4)
V_4 = Dilate(W_3, If[15 < a < 16, 16 - a, 1], C_1)
s_3 = Segment(W_3, V_4)
W_4 = Dilate(V_3, If[16 < a < 17, 17 - a, 1], D_1)
t_3 = Segment(V_3, W_4)
Z_4 = Dilate(T_3, If[17 < a < 18, 18 - a, 1], E_1)
a_4 = Segment(T_3, Z_4)
A_5 = Dilate(S_3, If[18 < a < 19, 19 - a, 1], F_1)
b_4 = Segment(S_3, A_5)
B_5 = Dilate(R_3, If[19 < a < 20, 20 - a, 1], G_1)
c_4 = Segment(R_3, B_5)
d_4 = If(4 < a < 5, Translate[p, (((0.8 * π) * (a - 4)), 0)])
C_5 = If(4 < a < 5, Translate[S_1, (((0.8 * π) * (a - 4)), 0)])
D_5 = Center(d_4)
E_5 = Rotate(C_5, ((4 - a) * 180°), D_5)
e_4 = CircleArc(D_5, I_4, E_5)
f_4 = If(5 < a < 6, Translate[q, (((0.75 * π) * (a - 5)), 0)])
F_5 = If(5 < a < 6, Translate[Q_2, (((0.75 * π) * (a - 5)), 0)])
G_5 = Center(f_4)
H_5 = Rotate(F_5, ((5 - a) * 180°), G_5)
g_4 = CircleArc(G_5, J_4, H_5)
h_4 = If(6 < a < 7, Translate[r, (((0.7 * π) * (a - 6)), 0)])
I_5 = If(6 < a < 7, Translate[D_3, (((0.7 * π) * (a - 6)), 0)])
J_5 = Center(h_4)
K_5 = Rotate(I_5, ((6 - a) * 180°), J_5)
k_4 = CircleArc(J_5, M_4, K_5)
p_4 = If(7 < a < 8, Translate[s, (((0.65 * π) * (a - 7)), 0)])
L_5 = If(7 < a < 8, Translate[E_3, (((0.65 * π) * (a - 7)), 0)])
M_5 = Center(p_4)
N_5 = Rotate(L_5, ((7 - a) * 180°), M_5)
q_4 = CircleArc(M_5, N_4, N_5)
r_4 = If(8 < a < 9, Translate[t, (((0.6 * π) * (a - 8)), 0)])
O_5 = If(8 < a < 9, Translate[F_3, (((0.6 * π) * (a - 8)), 0)])
P_5 = Center(r_4)
Q_5 = Rotate(O_5, ((8 - a) * 180°), P_5)
s_4 = CircleArc(P_5, O_4, Q_5)
t_4 = If(9 < a < 10, Translate[c_1, (((0.55 * π) * (a - 9)), 0)])
R_5 = If(9 < a < 10, Translate[G_3, (((0.55 * π) * (a - 9)), 0)])
S_5 = Center(t_4)
T_5 = Rotate(R_5, ((9 - a) * 180°), S_5)
c_5 = CircleArc(S_5, P_4, T_5)
d_5 = If(10 < a < 11, Translate[d_1, (((0.5 * π) * (a - 10)), 0)])
U_5 = If(10 < a < 11, Translate[H_3, (((0.5 * π) * (a - 10)), 0)])
V_5 = Center(d_5)
W_5 = Rotate(U_5, ((10 - a) * 180°), V_5)
e_5 = CircleArc(V_5, Q_4, W_5)
f_5 = If(11 < a < 12, Translate[e_1, (((0.45 * π) * (a - 11)), 0)])
Z_5 = If(11 < a < 12, Translate[I_3, (((0.45 * π) * (a - 11)), 0)])
A_6 = Center(f_5)
B_6 = Rotate(Z_5, ((11 - a) * 180°), A_6)
g_5 = CircleArc(A_6, R_4, B_6)
h_5 = If(12 < a < 13, Translate[f_1, (((0.4 * π) * (a - 12)), 0)])
C_6 = If(12 < a < 13, Translate[J_3, (((0.4 * π) * (a - 12)), 0)])
D_6 = Center(h_5)
E_6 = Rotate(C_6, ((12 - a) * 180°), D_6)
k_5 = CircleArc(D_6, S_4, E_6)
p_5 = If(13 < a < 14, Translate[g_1, (((0.35 * π) * (a - 13)), 0)])
F_6 = If(13 < a < 14, Translate[K_3, (((0.35 * π) * (a - 13)), 0)])
G_6 = Center(p_5)
H_6 = Rotate(F_6, ((13 - a) * 180°), G_6)
q_5 = CircleArc(G_6, T_4, H_6)
r_5 = If(14 < a < 15, Translate[h_1, (((0.3 * π) * (a - 14)), 0)])
I_6 = If(14 < a < 15, Translate[L_3, (((0.3 * π) * (a - 14)), 0)])
J_6 = Center(r_5)
K_6 = Rotate(I_6, ((14 - a) * 180°), J_6)
s_5 = CircleArc(J_6, U_4, K_6)
t_5 = If(15 < a < 16, Translate[k_1, (((0.25 * π) * (a - 15)), 0)])
L_6 = If(15 < a < 16, Translate[M_3, (((0.25 * π) * (a - 15)), 0)])
M_6 = Center(t_5)
N_6 = Rotate(L_6, ((15 - a) * 180°), M_6)
c_6 = CircleArc(M_6, V_4, N_6)
d_6 = If(16 < a < 17, Translate[p_1, (((0.2 * π) * (a - 16)), 0)])
O_6 = If(16 < a < 17, Translate[N_3, (((0.2 * π) * (a - 16)), 0)])
P_6 = Center(d_6)
Q_6 = Rotate(O_6, ((16 - a) * 180°), P_6)
e_6 = CircleArc(P_6, W_4, Q_6)
f_6 = If(17 < a < 18, Translate[q_1, (((0.15 * π) * (a - 17)), 0)])
R_6 = If(17 < a < 18, Translate[O_3, (((0.15 * π) * (a - 17)), 0)])
S_6 = Center(f_6)
T_6 = Rotate(R_6, ((17 - a) * 180°), S_6)
g_6 = CircleArc(S_6, Z_4, T_6)
h_6 = If(18 < a < 19, Translate[r_1, (((0.1 * π) * (a - 18)), 0)])
U_6 = If(18 < a < 19, Translate[P_3, (((0.1 * π) * (a - 18)), 0)])
V_6 = Center(h_6)
W_6 = Rotate(U_6, ((18 - a) * 180°), V_6)
k_6 = CircleArc(V_6, A_5, W_6)
p_6 = If(19 < a < 20, Translate[s_1, (((0.05 * π) * (a - 19)), 0)])
Z_6 = If(19 < a < 20, Translate[Q_3, (((0.05 * π) * (a - 19)), 0)])
A_7 = Center(p_6)
B_7 = Rotate(Z_6, ((19 - a) * 180°), A_7)
q_6 = CircleArc(A_7, B_5, B_7)
r_6 = Mirror(e_4, f)
C_7 = Mirror(R_2, f)
D_7 = Mirror(J_4, f)
i_4 = Segment(C_7, D_7)
s_6 = Mirror(g_4, f)
t_6 = Mirror(k_4, f)
E_7 = Mirror(H_4, f)
F_7 = Mirror(M_4, f)
j_4 = Segment(E_7, F_7)
G_7 = Mirror(G_4, f)
H_7 = Mirror(N_4, f)
l_4 = Segment(G_7, H_7)
c_7 = Mirror(q_4, f)
I_7 = Mirror(F_4, f)
J_7 = Mirror(O_4, f)
m_4 = Segment(I_7, J_7)
d_7 = Mirror(s_4, f)
K_7 = Mirror(E_4, f)
L_7 = Mirror(P_4, f)
n_4 = Segment(K_7, L_7)
e_7 = Mirror(c_5, f)
M_7 = Mirror(D_4, f)
N_7 = Mirror(Q_4, f)
a_5 = Segment(M_7, N_7)
f_7 = Mirror(e_5, f)
O_7 = Mirror(C_4, f)
P_7 = Mirror(R_4, f)
b_5 = Segment(O_7, P_7)
g_7 = Mirror(g_5, f)
Q_7 = Mirror(B_4, f)
R_7 = Mirror(S_4, f)
i_5 = Segment(Q_7, R_7)
h_7 = Mirror(k_5, f)
S_7 = Mirror(A_4, f)
T_7 = Mirror(T_4, f)
j_5 = Segment(S_7, T_7)
k_7 = Mirror(q_5, f)
U_7 = Mirror(Z_3, f)
V_7 = Mirror(U_4, f)
l_5 = Segment(U_7, V_7)
p_7 = Mirror(s_5, f)
W_7 = Mirror(W_3, f)
Z_7 = Mirror(V_4, f)
m_5 = Segment(W_7, Z_7)
q_7 = Mirror(c_6, f)
A_8 = Mirror(V_3, f)
B_8 = Mirror(W_4, f)
n_5 = Segment(A_8, B_8)
r_7 = Mirror(e_6, f)
C_8 = Mirror(T_3, f)
D_8 = Mirror(Z_4, f)
a_6 = Segment(C_8, D_8)
s_7 = Mirror(g_6, f)
E_8 = Mirror(S_3, f)
F_8 = Mirror(A_5, f)
b_6 = Segment(E_8, F_8)
t_7 = Mirror(k_6, f)
G_8 = Mirror(R_3, f)
H_8 = Mirror(B_5, f)
i_6 = Segment(G_8, H_8)
c_8 = Mirror(q_6, f)
i = Segment((0, 0), (0, 1))
j_6 = Segment(A, F)
l_6 = Segment(F, B)
e_8 = Circle((-6, 1), 1)
a_7 = Segment((-6, 1), (-6, 0))
I_8 = (-6, 1)