Area of a circle
Author: Malin Christersson
Date: 19 March 2013

Demonstration.

r

r\cdot \pi

$ × 4 $

t2 = 1 (slider from 0 to 1, step 0.01)
n = 3 (slider from 1 to 30, step 1)
d: sqrt(2 - (2 * cos(π / ((2 * n)))))
t1 = 1 (slider from 0 to 1, step 0.01)
a: (t1 * π / ((4 * n)))
A1 = 0
A2: A1 + 1
A3: A2 + 1
A4: A3 + 1
A5: A4 + 1
A6: A5 + 1
A7: A6 + 1
A8: A7 + 1
A9: A8 + 1
A10: A9 + 1
A11: A10 + 1
A12: A11 + 1
A13: A12 + 1
A14: A13 + 1
A15: A14 + 1
A16: A15 + 1
A17: A16 + 1
A18: A17 + 1
A19: A18 + 1
A20: A19 + 1
A21: A20 + 1
A22: A21 + 1
A23: A22 + 1
A24: A23 + 1
A25: A24 + 1
A26: A25 + 1
A27: A26 + 1
A28: A27 + 1
A29: A28 + 1
A30: A29 + 1
A31: A30 + 1
angles = Sequence((((2 * i) - 1) * a), i, 1, n)
vectors = Sequence(Vector[(d * Vector[(cos((-Element[angles, i])), sin((-Element[angles, i])))])], i, 1, n)
A: (0, 1) + ((1 - t2) * Vector[(0, 1), Mirror[Intersect[Circle[(0, 1), 1], OrthogonalLine[Midpoint[(0, 1), (0, 1) + (d * Vector[(cos((-a)), sin((-a)))])], Line[(0, 1), (0, 1) + (d * Vector[(cos((-a)), sin((-a)))])]]], xAxis]])
B1: A
B2: B1 + Element[vectors, A2]
B3: B2 + Element[vectors, A3]
B4: B3 + Element[vectors, A4]
B5: B4 + Element[vectors, A5]
B6: B5 + Element[vectors, A6]
B7: B6 + Element[vectors, A7]
B8: B7 + Element[vectors, A8]
B9: B8 + Element[vectors, A9]
B10: B9 + Element[vectors, A10]
B11: B10 + Element[vectors, A11]
B12: B11 + Element[vectors, A12]
B13: B12 + Element[vectors, A13]
B14: B13 + Element[vectors, A14]
B15: B14 + Element[vectors, A15]
B16: B15 + Element[vectors, A16]
B17: B16 + Element[vectors, A17]
B18: B17 + Element[vectors, A18]
B19: B18 + Element[vectors, A19]
B20: B19 + Element[vectors, A20]
B21: B20 + Element[vectors, A21]
B22: B21 + Element[vectors, A22]
B23: B22 + Element[vectors, A23]
B24: B23 + Element[vectors, A24]
B25: B24 + Element[vectors, A25]
B26: B25 + Element[vectors, A26]
B27: B26 + Element[vectors, A27]
B28: B27 + Element[vectors, A28]
B29: B28 + Element[vectors, A29]
B30: B29 + Element[vectors, A30]
B31: B30 + Element[vectors, A31]
midpoints = Sequence(Midpoint[Cell[2, i], Cell[2, 1 + i]], i, 1, n)
lowpoints = Sequence(Intersect[Circle[Cell[2, i], 1], OrthogonalLine[Element[midpoints, i], Line[Cell[2, i], Cell[2, 1 + i]]]], i, 1, n)
lowpoints2 = Sequence(Mirror[Element[lowpoints, i], yAxis], i, 1, n)
C1 = (0, -1)
C2: C1 - Element[vectors, A2]
C3: C2 - Element[vectors, A3]
C4: C3 - Element[vectors, A4]
C5: C4 - Element[vectors, A5]
C6: C5 - Element[vectors, A6]
C7: C6 - Element[vectors, A7]
C8: C7 - Element[vectors, A8]
C9: C8 - Element[vectors, A9]
C10: C9 - Element[vectors, A10]
C11: C10 - Element[vectors, A11]
C12: C11 - Element[vectors, A12]
C13: C12 - Element[vectors, A13]
C14: C13 - Element[vectors, A14]
C15: C14 - Element[vectors, A15]
C16: C15 - Element[vectors, A16]
C17: C16 - Element[vectors, A17]
C18: C17 - Element[vectors, A18]
C19: C18 - Element[vectors, A19]
C20: C19 - Element[vectors, A20]
C21: C20 - Element[vectors, A21]
C22: C21 - Element[vectors, A22]
C23: C22 - Element[vectors, A23]
C24: C23 - Element[vectors, A24]
C25: C24 - Element[vectors, A25]
C26: C25 - Element[vectors, A26]
C27: C26 - Element[vectors, A27]
C28: C27 - Element[vectors, A28]
C29: C28 - Element[vectors, A29]
C30: C29 - Element[vectors, A30]
C31: C30 - Element[vectors, A31]
midpointsDown = Sequence(Midpoint[Cell[3, i], Cell[3, 1 + i]], i, 1, n)
uppoints = Sequence(Intersect[Circle[Cell[3, i], 1], OrthogonalLine[Element[midpointsDown, i], Line[Cell[3, i], Cell[3, 1 + i]]]], i, 1, n)
uppoints2 = Sequence(Mirror[Element[uppoints, i], yAxis], i, 1, n)
B = Point(yAxis)
g = OrthogonalLine(B, yAxis)
C = Point(g)
C' = Mirror(C, yAxis)
D = Point(h)
D' = Mirror(D, (-2, -0.5))
E = Intersect(b, xAxis)
F = Intersect(c, h)
G = Intersect(e, g)
H = Intersect(g, f)
I = (-2, 0)
u = Vector(D', I)
v = Vector(D, F)
w = Vector(C', G)
z = Vector(C, H)
sectorsDown1 = Sequence(CircleSector[Element[uppoints, i], Cell[3, 1 + i], Cell[3, i]], i, 1, n, 2)
sectorsDown3 = Sequence(CircleSector[Element[uppoints2, i], Mirror[Cell[3, i], yAxis], Mirror[Cell[3, 1 + i], yAxis]], i, 1, n, 2)
sectors1 = Sequence(CircleSector[Element[lowpoints, i], Cell[2, 1 + i], Cell[2, i]], i, 1, n, 2)
sectors3 = Sequence(CircleSector[Element[lowpoints2, i] + ((2 * x(A)), 0), Mirror[Cell[2, i], yAxis] + ((2 * x(A)), 0), Mirror[Cell[2, 1 + i], yAxis] + ((2 * x(A)), 0)], i, 1, n, 2)
sectors2 = Sequence(CircleSector[Element[lowpoints, i], Cell[2, 1 + i], Cell[2, i]], i, 2, n, 2)
sectors4 = Sequence(CircleSector[Element[lowpoints2, i] + ((2 * x(A)), 0), Mirror[Cell[2, i], yAxis] + ((2 * x(A)), 0), Mirror[Cell[2, 1 + i], yAxis] + ((2 * x(A)), 0)], i, 2, n, 2)
sectorsDown2 = Sequence(CircleSector[Element[uppoints, i], Cell[3, 1 + i], Cell[3, i]], i, 2, n, 2)
sectorsDown4 = Sequence(CircleSector[Element[uppoints2, i], Mirror[Cell[3, i], yAxis], Mirror[Cell[3, 1 + i], yAxis]], i, 2, n, 2)
m: (4 * n)
text4: "" + m + " Sectors"