Download Math 7030 - UGA Math Department

Math 7030
Activity 3 - The Five Pointed Star
In this activity you will see the connection between the Five Pointed Star and the Golden
Ratio. We will say that an isosceles triangle is golden if the equal sides are longer than
the remaining side and the ratio of one of the longer sides to the shorter side is  , the
golden ratio. We know that for an isosceles triangle is golden if and only if the angles are
72o, 72o, and 36o.
C
D
O
N
P
A
B
R
Q
E
1.
2.
3.
4.
5.
6.
7.
F
Construct a five-pointed star using GSP by taking your regular pentagon and
constructing the diagonals.
Assume the edge length of the regular pentagon BCDEF is 1.
Prove that angle DCB = 108o.
Prove that angle NCB = 36o.
Prove that angle OCN = 36o.
Prove that triangle ECF is golden and therefore EC =  .
1
Prove that triangle OCB is golden and therefore OC  .

8. Prove that triangle OCN is golden and therefore ON 
1
2
This is an interesting result as it gives the ratio of the edge length of the smaller
pentagon to the edge length of the original pentagon.
You are now going to calculate the area of the original pentagon. To do this we will
assume Heron's formula for the area of a triangle having sides with lengths a, b and c.
abc
We let s 
. Then the area A of the triangle is given by
2
A  s(s  a)(s  b)(s  c)
We break the pentagon up into three disjoint triangles EDC, ECF, and FCB. By
symmetry the area of triangles EDC and FCB are equal If we let A1 denote the area of
triangle EDC and A2 denote the area of triangle ECF, then the area A of the pentagon
is given by A = 2A1 + A2.
9. Prove that A1 

2
1
2
4
.
1 2 1
 
2
4
11. Show that the area of the original pentagon is approximately 1.7204.
12. The above result is for a pentagon having edge length 1. What is the area of a
pentagon whose edge length is some positive number r? You can check your
formula by having GSP measure the edge length and area of your original
pentagon.
10. Prove that A2 