Area of a Rhombus
Author: Barbara Perez
Date: 1/30/07

Use this worksheet to derive the formula for area of a rhombus

Use the slider to change the side lengths of the rhombus.  Use point C to change the angles.
Use the sliders to reflect all four triangles.
1. What shape does the rhombus and the four triangles form?
2. What is the length of the sides of the combined figure in terms of d1 and d2?
3. Write a formula for the area of the combined figure in terms of d1 and d2.
4. Write a formula for the area of rhombus ABCD in terms of d1 and d2. (Hint: What is the relationship of the area of the rhombus to the area of the new figure?)

f

wielkość

obrót Δ₄

obrót Δ₃

obrót Δ₂

obrót Δ₁

POLE ROMBU

Pole =\frac{\textcolor{204,0,51}{ e} \cdot \textcolor{0,50,153}{ f }}{2} 

e

wymiary

f

e

pokaż wzór

a = 5 (slider from 1 to 5, step 1)
A = (-2.8, 4.8)
B = Point(Circle[A, a])
b = Segment(A, B)
c = Circle(A, a)
C = Point(c)
d = Segment(A, C)
e = Circle(C, a)
f = Line(C, b)
D = Intersect(e, f, 2)
g = Segment(C, D)
h = Segment(D, B)
m = Segment(A, D)
n = Segment(C, B)
E = Intersect(m, n)
P, d_1, c_1, e_1 = Polygon(C, E, D)
Q, a_1, c_2, e_2 = Polygon(C, E, A)
R, b_1, a_2, e_3 = Polygon(A, E, B)
S, e_4, b_2, d_2 = Polygon(B, D, E)
F = Midpoint(D, C)
G = Midpoint(D, B)
H = Midpoint(A, B)
I = Midpoint(A, C)
C' = Mirror(C, I)
E' = Mirror(E, I)
A' = Mirror(A, I)
C'_1 = Mirror(C, F)
E'_1 = Mirror(E, F)
D' = Mirror(D, F)
B' = Mirror(B, G)
D'_1 = Mirror(D, G)
E'_2 = Mirror(E, G)
A'_1 = Mirror(A, H)
E'_3 = Mirror(E, H)
B'_1 = Mirror(B, H)
α = 0 rad (slider from 0 to 3.141592653589793, step 0.017453292519943295)
β = 0 rad (slider from 0 to 3.141592653589793, step 0.017453292519943295)
γ = 0 rad (slider from 0 to 3.141592653589793, step 0.017453292519943295)
δ = 0 rad (slider from 0 to 3.141592653589793, step 0.017453292519943295)
C'_2 = Rotate(C, α, I)
E'_4 = Rotate(E, α, I)
A'_2 = Rotate(A, α, I)
Q', a'_2, c'_2, e'_4 = Polygon(C'_2, E'_4, A'_2)
C'_3 = Rotate(C, -β, F)
E'_5 = Rotate(E, -β, F)
D'_2 = Rotate(D, -β, F)
P', d'_2, c'_3, e'_5 = Polygon(C'_3, E'_5, D'_2)
A'_3 = Rotate(A, γ, H)
E'_6 = Rotate(E, γ, H)
B'_2 = Rotate(B, γ, H)
R', b'_2, a'_3, e'_6 = Polygon(A'_3, E'_6, B'_2)
B'_3 = Rotate(B, -δ, G)
D'_3 = Rotate(D, -δ, G)
E'_7 = Rotate(E, -δ, G)
S', e'_7, b'_3, d'_3 = Polygon(B'_3, D'_3, E'_7)
i = Line(E'_2, E'_3)
j = OrthogonalLine(E'_2, i)
k = OrthogonalLine(E'_3, i)
l = OrthogonalLine(E'_1, j)
J = Point(i)
p = OrthogonalLine(J, i)
K = Point(j)
q = OrthogonalLine(K, j)
L = Intersect(k, q)
M = Intersect(l, p)
u = Vector(K, L)
v = Vector(L, K)
w = Vector(M, J)
z = Vector(J, M)
N = Point(j)
O = Point(k)
T = Point(i)
U = Point(l)
r = Segment(D, N)
s = Segment(A, O)
t = Segment(B, T)
f_1 = Segment(C, U)
g_1 = Segment(B, E'_3)
h_1 = Segment(D, E'_1)
i_1 = Segment(C, E')
j_1 = Segment(E', A)
o = false
k_1 = Segment(K, L)
l_1 = Segment(J, M)
V = Midpoint(k_1)
W = Midpoint(l_1)
m_1 = true