Regular Pentagons and the Golden Ratio

Name ____________________ Hour _____
REGULAR PENTAGONS and
THE GOLDEN RATIO
Pictured below is a REGULAR PENTAGON. In it, there are a number of segments of different
lengths. Measure the ones indicated below to the nearest 0.5 mm.
1. List the lengths of the following segments (to the nearest .5 mm):
a. EF=EJ=JD=DI=IC=CH=HB=BG=GA=AF=
b. FJ=JI=IH=HG=GF=
c. EC=EB=DA=DB=CE=CA=BD=BE=AC=AD=
d. ED=DC=CB=BA=AE=
Revised August 3, 2000
Steve Boast

REGULAR PENTAGONS and
THE GOLDEN RATIO
2. Certain combinations of these segments form congruent, isosceles triangles. List all such
triangles below.
a. The smaller congruent isosceles triangles are:
b. The larger congruent isosceles triangles are:
3. Find the ratio of the length of the side and the length of the base of each size triangle.
Smaller Isosceles Triangle Larger Isosceles Triangle
Side Side
)))) = )))) =
Base Base
4. If a triangle has two sides whose ratio of lengths is the GOLDEN RATIO, the triangle is
called a GOLDEN TRIANGLE. Which of the 10 triangles you listed are GOLDEN
TRIANGLES?
1+
5
Hint: The GOLDEN RATIO is equal to .
2
5. What is the measure of the vertex angle and the base angles in each of the smaller isosceles
triangles? Explain
Revised August 3, 2000
Steve Boast Page 2

REGULAR PENTAGONS and
THE GOLDEN RATIO
6. What is the measure of the vertex angle and the base angles in each of the larger isosceles
triangles? Explain
7. Are the smaller isosceles triangles similar to the larger isosceles triangles? Why?
8. Are GOLDEN TRIANGLES always isosceles? Explain
9. Are all GOLDEN TRIANGLES similar? Explain
Revised August 3, 2000
Steve Boast Page 3