AreaCicle
Author: Daniel Mentrard
Date: 6 Mai 2013

circumference = 2 π r

The area of a circle

r

A= \frac{h *B}{2}

h

B

What is the area of this triangle?

R = 1.1 (slider from 0 to 5, step 0.1)
A: (0, R + 0.002)
O: A + (0, R)
u = Vector(O, A)
Extend = 0.52 (slider from 0 to 1, step 1.0E-4)
Q: O + Extend * u
P: Q + (0, -R)
m = 14 (slider from 1 to 50, step 1)
Animat1 = Min(Max[betaRegularized(2, 2.3, Extend), 1.0E-4], 0.9999)
f(x) = tan(x * π / 2)
Animat2: f(Animat1)
B: A + (0, -R)
C: A + (0, Animat2)
Pointdebut = Sequence(A + (0, -i), i, R / m, R, R / m)
Cercles = Zip(Circle[C, B], B, Pointdebut)
PointFins = Zip(Rotate[B, 180 * ° / (2 * Radius[K] * π) * 2 * Distance[A, B] * π, C], K, Cercles, B, Pointdebut)
B' = Element(PointFins, Length[PointFins])
b = Segment(B, B')
D = Midpoint(B, B')
c = OrthogonalLine(D, b)
ABS = PolyLine(Append[A, PointFins])
K = Intersect(c, ABS)
a = Circle(A, R)
Arcs = Zip(CircleArc[C, B, B'], B, Pointdebut, B', PointFins)
g = Segment(A, B')
texte3: "dissect the circle into " + m + "  parts"
aa = Element(Arcs, m)
h = Segment(Q, P)
E = Midpoint(h)
F = (-4.832830355033505, 0.6260515764285839)
liste2 = Zip(CircleArc[C, B, B'], B, Pointdebut, B', PointFins / 2)
liste3 = Sequence(Intersect[Element[Arcs, i], c], i, 1, Length[Arcs])
L = Intersect(aa, b, 2)
poly3, k_1, d_1, l = Polygon(D, L, K)
M = Point(h)
i = Segment(M, D)
PointFins2 = Zip(Rotate[B, -180 * ° / (2 * Radius[K] * π) * 2 * Distance[A, B] * π, C], K, Cercles, B, Pointdebut)
N = Element(PointFins2, Length[PointFins2])
r = CircleArc(C, N, B)
Arcs2 = Zip(CircleArc[C, N, B], N, PointFins2, B, Pointdebut)
bb = Element(Arcs2, m)
j = PolyLine(Append[A, PointFins2])
H: F + (0, 1)
G: F + (1, 0)
I: F + (-1, 0)
poly1, h_1, i_1, g_1 = Polygon(I, G, H)
J = Intersect(g_1, i_1)
d = Segment(J, F)
S = Midpoint(d)