Unrolling a circle
Date: 5 April 2013

When a bike tire turns one revolution the bike travels a distance equal to the circumference of the tire. Here is a visualtion to help students understand why.

roll

0

0

1

2π

1

2

3

π

4

5

6

Watch what happens as the circle unrolls. 
Notice that each part of the arc has a length equal to the radius. 
What is the length of the whole circumference?
Explain what the protractor measures. 

Rolling a Unit Circle

1

1

1

1

1

1

0

π

2π

0°

90°

180°

270°

45°

135°

225°

315°

15°

30°

60°

75°

105°

120°

150°

165°

195°

240°

255°

285°

300°

330°

345°

210°

center = -3.99 (slider from -3.99 to 8.56, step 0.0523)
A = (-4, -1)
B = (15, -1)
a = Line(A, B)
C = (-7, 1)
D = (-6, 1)
b = Line(C, D)
E: (center, 1)
c = Circle(E, 2)
d = OrthogonalLine(E, a)
F = Intersect(d, a)
e = Segment(A, F)
F' = Rotate(F, (-(180 * e / 6.28))°, E)
α = Angle(F', E, F)
f = CircleArc(E, F, F')
u = Vector(E, F')
G = (8.56, -1)
H = (-2.01, -1)
I = (-0.02, -1)
F''_1 = Rotate(F', 3.141592653589793, E)
K = (1.98, -1)
L = (2.28, -1)
M = (4, -1)
N = (6, -1)
O_1 = (8, -1)
P = (-3.99, -1.5)
Q = (-1.99, -1.5)
v = Vector(P, Q)
Q' = Translate(Q, v)
w = Vector(Q, Q')
Q'' = Translate(Q', w)
Q''' = Translate(Q'', w)
R = Translate(Q''', w)
R' = Translate(R, w)
z = Vector(Q', Q'')
m = Vector(Q'', Q''')
n = Vector(Q''', R)
p = Vector(R, R')
q = Vector(P, R')
J = (-4, -3.96)
S = (-4.033057851239669, -2.4165289256198346)
F'' = Rotate(F', 0.7853981633974483, E)
β = Angle(F', E, F'')
F''_2 = Rotate(F', 0.2617993877991494, E)
γ = Angle(F', E, F''_2)
F''_3 = Rotate(F', 0.5235987755982988, E)
δ = Angle(F', E, F''_3)
F''_4 = Rotate(F', 1.0471975511965976, E)
ε = Angle(F', E, F''_4)
F''_5 = Rotate(F', 1.3089969389957472, E)
ζ = Angle(F', E, F''_5)
F''_6 = Rotate(F', 1.5707963267948966, E)
η = Angle(F', E, F''_6)
F''_7 = Rotate(F', 1.8325957145940461, E)
θ = Angle(F', E, F''_7)
F''_8 = Rotate(F', 2.0943951023931953, E)
ι = Angle(F', E, F''_8)
U = Rotate(F''_8, 0.2617993877991494, E)
κ = Angle(F''_8, E, U)
U' = Rotate(U, 0.2617993877991494, E)
λ = Angle(U, E, U')
U'' = Rotate(U', 0.2617993877991494, E)
μ = Angle(U', E, U'')
U''' = Rotate(U'', 0.2617993877991494, E)
ν = Angle(U'', E, U''')
V = Rotate(U''', 0.2617993877991494, E)
ξ = Angle(U''', E, V)
V' = Rotate(V, 0.2617993877991494, E)
ο = Angle(V, E, V')
V'' = Rotate(V', 0.2617993877991494, E)
ρ = Angle(V', E, V'')
V''' = Rotate(V'', 0.2617993877991494, E)
σ = Angle(V'', E, V''')
W = Rotate(V''', 0.2617993877991494, E)
τ = Angle(V''', E, W)
W' = Rotate(W, 0.2617993877991494, E)
υ = Angle(W, E, W')
W'' = Rotate(W', 0.2617993877991494, E)
ϕ = Angle(W', E, W'')
W''' = Rotate(W'', 0.2617993877991494, E)
χ = Angle(W'', E, W''')
Z = Rotate(W''', 0.2617993877991494, E)
ψ = Angle(W''', E, Z)
Z' = Rotate(Z, 0.2617993877991494, E)
ω = Angle(Z, E, Z')
Z'' = Rotate(Z', 0.2617993877991494, E)
α_1 = Angle(Z', E, Z'')
T = (8.95, -3.96)
A_1 = (-4, -3.37)
B_1 = Mirror(A_1, J)
g = Line(B_1, a)
h = OrthogonalLine(T, g)
C_1 = Intersect(g, h)
i = Segment(J, B_1)
j = Segment(B_1, C_1)
k = Segment(C_1, T)
l = Segment(T, J)
D_1 = Point(l)
r = OrthogonalLine(D_1, j)
E_1 = Intersect(j, r)
s = Segment(D_1, E_1)
t = OrthogonalLine(G, a)
F_1 = Intersect(t, j)
G_1 = Intersect(t, l)
a_1 = Segment(F_1, G_1)
poly1, b_1, j_1, t_1, c_1 = Polygon(B_1, J, T, C_1)
d_1 = OrthogonalLine(E, a)
H_1 = Point(d_1)
e_1 = Segment(H_1, F''_2)
f_1 = Segment(H_1, F''_3)
g_1 = Segment(H_1, F'')
h_1 = Segment(H_1, F''_4)
i_1 = Segment(H_1, F''_5)
k_1 = Segment(H_1, F''_6)
l_1 = Segment(H_1, F''_7)
m_1 = Segment(H_1, F''_8)
n_1 = Segment(H_1, U)
p_1 = Segment(H_1, U')
q_1 = Segment(H_1, U'')
I_1 = Point(d_1)
r_1 = Segment(H_1, I_1)
s_1 = Segment(H_1, V)
a_2 = Segment(H_1, V')
b_2 = Segment(H_1, V'')
c_2 = Segment(H_1, V''')
d_2 = Segment(H_1, W)
e_2 = Segment(H_1, W')
f_2 = Segment(H_1, W'')
g_2 = Segment(H_1, W''')
h_2 = Segment(H_1, Z)
i_2 = Segment(H_1, Z')
j_2 = Segment(H_1, Z'')
J_1 = Point(d_1)
k_2 = Ray(H_1, J_1)
K_1 = Point(k_2)
l_2 = Segment(H_1, K_1)
m_2 = Segment(F', H_1)
n_2 = Segment(H_1, U''')
p_2 = Ray(H_1, K_1)
O = Intersect(e_1, f_1)
L_1 = Point(m_2)
M_1 = Point(k_1)
N_1 = Point(n_2)
P_1 = Point(e_2)
Q_1 = Point(g_1)
R_1 = Point(n_1)
S_1 = Point(b_2)
T_1 = Point(h_2)
U_1 = Point(e_1)
V_1 = Point(f_1)
W_1 = Point(h_1)
Z_1 = Point(i_1)
A_2 = Point(l_1)
B_2 = Point(m_1)
C_2 = Point(p_1)
D_2 = Point(q_1)
E_2 = Point(s_1)
F_2 = Point(c_2)
G_2 = Point(d_2)
H_2 = Point(f_2)
I_2 = Point(g_2)
J_2 = Point(i_2)
K_2 = Point(j_2)
L_2 = Point(a_2)