Nombre de côtés

Animation

Montrer les sommets

A

B

C

D

Exibir dica 1

\overline{AB} \mbox{ tend vers } \pi \cdot r, \mbox{ quand } k \mbox{ tend vers }  +\infty.

H

\overline{DH} \mbox{ tend vers } r \mbox{ quand } k \mbox{ tend vers }  +\infty.

Exibir dica 2

Afficher l'explication

\mbox{ Aire de } ABCD = \overline{AB} \cdot \overline{DH} \mbox{ tend vers } (\pi \cdot r) \cdot r = \pi \cdot r^{2} = \mbox{ aire du cercle } \mbox{de rayon r, quand } \k \mbox{ tend vers }  +\infty.

(en fin d'animation)

Aire d'un disque

D'après UFF (Universidade Federal Fluminense)

r = 2
l: 3.141592653589793 * r
P: (0, -r)
k = 2 (slider from 1 to 40, step 1)
n: 2 * k
t = 0 (slider from 0 to 2, step 0.01)
tf = Max(t - 1, 0)
v = Min(t, 0.9999)
O: (0, tan(v * 3.141592653589793 / 2))
O' = Mirror(O, a)
R: r + y(O)
alpha: l / (R * n)
AB = Rotate(P, (-l) / (2 * R), O)
AB' = Mirror(AB, a)
alist = Sequence(Rotate[AB', -i * alpha, O'], i, 0, n)
h: r * cos(3.141592653589793 / (2 * n))
s: 2 * r * sin(3.141592653589793 / (2 * n))
H: ((-h) / s * (y(Element[alist, 2]) - y(Element[alist, 1])), h / s * (x(Element[alist, 2]) - x(Element[alist, 1])))
hreal: y(H)
tv: (tf / 4 * (x(Element[alist, 2]) - x(Element[alist, 1])), tf * (r - hreal / 2))
blist = Sequence(Rotate[AB, i * alpha, O], i, 0, n)
ahlist = Sequence((x(Element[alist, i]) - x(tv), y(Element[alist, i]) - y(tv)), i, 1, n + 1)
bhlist = Sequence((x(Element[blist, i]) + x(tv), y(Element[blist, i]) + y(tv)), i, 1, n + 1)
P' = Mirror(P, a)
bmlist = Sequence(Midpoint[Element[bhlist, i], Element[bhlist, i + 1]], i, 1, n)
bnlist = Sequence(((-h) / s * (y(Element[bhlist, i + 1]) - y(Element[bhlist, i])), h / s * (x(Element[bhlist, i + 1]) - x(Element[bhlist, i]))), i, 1, n)
amlist = Sequence(Midpoint[Element[ahlist, i], Element[ahlist, i + 1]], i, 1, n)
bvlist = Sequence((x(Element[bmlist, i]) + x(Element[bnlist, i]), y(Element[bmlist, i]) + y(Element[bnlist, i])), i, 1, n)
anlist = Sequence(((-h) / s * (y(Element[ahlist, i + 1]) - y(Element[ahlist, i])), h / s * (x(Element[ahlist, i + 1]) - x(Element[ahlist, i]))), i, 1, n)
avlist = Sequence((x(Element[amlist, i]) - x(Element[anlist, i]), y(Element[amlist, i]) - y(Element[anlist, i])), i, 1, n)
sa1 = Segment((0, 0), (0, r))
sa2 = Segment((0, 0), (0, -r))
sa3 = Segment((r, 0), (0, 0))
sa4 = Segment((-r, 0), (0, 0))
aclist = Sequence(CircleSector[Element[avlist, i], Element[ahlist, i + 1], Element[ahlist, i]], i, 1, n)
bclist = Sequence(CircleSector[Element[bvlist, i], Element[bhlist, i], Element[bhlist, i + 1]], i, 1, n)
bplist = Sequence(Polygon[Element[bhlist, i], Element[bhlist, i + 1], Element[bvlist, i]], i, 1, n)
aplist = Sequence(Polygon[Element[ahlist, i], Element[ahlist, i + 1], Element[avlist, i]], i, 1, n)
showVertices = false
PA = Element(avlist, 1)
PB = Element(bhlist, n + 1)
PC = Element(bvlist, n)
PD = Element(ahlist, 1)
par, ra, rb, rc, rd = Polygon(PA, PB, PC, PD)
sPAPB = Segment(PA, PB)
sPBPC = Segment(PB, PC)
sPCPD = Segment(PC, PD)
sPDPA = Segment(PD, PA)
showHint1 = false
b = Line(PA, PB)
c = OrthogonalLine(PD, b)
PH = Intersect(b, c)
sPAPH = Segment(PA, PH)
sPDPH = Segment(PH, PD)
α = Angle(PB, PH, PD)
showHint2 = false
list1 = TextToUnicode("Área")
list2 = TextToUnicode("área do círculo")
list3 = TextToUnicode("área do círculo")
list4 = TextToUnicode("área do círculo")
list5 = TextToUnicode("área do círculo")
showHint3 = false