Defining the Golden Ratio
1. You and a partner should pick up the worksheet titled “Where Are
the Golden Rectangles?” and a ruler from your teacher. Before
beginning step 2, you and your partner should pick one rectangle
from each set that you like the best, or find the most pleasing.
2. Measure the length and width of each rectangle to the nearest
millimeter. You can write the results directly on the worksheet.
3. One person should open Spreadsheet 1, while the other reads the
directions on the next page.
Spreadsheet 1
Completing Spreadsheet 1: Finding the Golden Rectangle
•Step 1) Fill in the length of each rectangle in Column B.
•Step 2) Fill in the width of each rectangle in Column C.
•Step 3) In cell D2, type in the formula:
=B2/C2
•Step 4) Click on cell D2 again. Under the Edit menu at the top of the
screen, choose Copy. Your cell is now outlined.
•Step 5) Click on cell D3. Holding your left mouse key, drag down the
remainder of column D (through row 13). The column should
now be shaded black.
•Step 6) Under the Edit menu, choose Paste. What has happened in each
of the cells? Was this expected?
•Step 7) Repeat steps 4-6 for the E column. Your formula for cell E2
should be: =C2/B2
Defining the Golden Ratio
Looking at Spreadsheet 1, which rectangles in column D have the
same value? What about column E?
Rectangles B, H, and J are called Golden Rectangles because their
ratio of length to width (or width to length) approximates the
Golden Ratio. (Were these the rectangles that you and your partner
preferred from each set?)
The Golden Ratio is an irrational number (like Pi) and is denoted by
the Greek letter, Phi. When dividing the length by the width, the
approximate value of the golden ratio is 1.618. When the width is
divided by the length, the value is approximately 0.618.
Now that we’ve defined the golden ratio, we are going to look at two
of its applications.
(Your partner should close Spreadsheet 1 by clicking on the “x” at
the very top right corner of the screen and proceed to this slide.)
Application 1: Architecture
The ancient Greeks believed that the Golden Rectangle was one of
the most pleasing shapes to the eye. Thus, many of their buildings,
such as the Parthenon (pictured above center) are composed of
golden rectangles. Likewise, many modern works of architecture,
such as the United Nations Headquarters, are also made up of
golden rectangles.
For our next project, we are going to explore the golden rectangles
that make up the Parthenon in a little more detail.
Application 1: Architecture
Pick up the worksheet titled “Find the Parthenon’s Golden Rectangles”
from your teacher. Your job is to outline as many golden rectangles as
you can find in the architecture of the Parthenon. To do this, you will
want to measure the length and width of each rectangle and then find
the ratio of length divided by width.
To make this process go more quickly, you may want to type the
lengths and widths into a spreadsheet and then calculate the ratio
using a formula. I’ve started a spreadsheet for you. You just need to
type in the appropriate formula in column C.
See Spreadsheet 2.
How many golden rectangles did you find?
Application 2: Art
In addition to architecture, the golden ratio can be found in many
places in the human body. When one measures various body parts in
relation to the whole (such as the length measured from the top of
the head to the floor divided by the length measured from the navel
to the floor), a golden ratio is found. Mathematically, the ratio looks
like this:
AC AB

AB BC
A
B
C
Many artists are aware of this fact and use mathematics and the
golden ratio when planning the composition of their paintings
(or sculptures) to make the work seem more realistic.
Application 2: Art
For our last project, we are going to explore the golden ratios that
occur in human beings.
•Pick up the worksheet titled “My Body Measurements” and a tape
measure from the teacher. You and your partner should measure
each other and fill in sections A and B for each part on the Data
Recording Sheet. You will then use a spreadsheet to calculate the
remaining pieces.
•Use the metric system when choosing your units.
•Please note that the units used for the entire height of the person
(such as meters or centimeters) do not have to be the same as those
used to measure the parts of the finger (millimeters). Just make
sure that you are using a common measure for each separate part.
•Once you have filled out sections A and B, type in the appropriate
formulas to complete the spreadsheet below. (There are two areas to
complete, one for your measurements and one for your partner’s.)
Spreadsheet 3
Application 2: Art
Did you and your partner discover any golden ratios?
In which parts were your measurements pretty close to the golden
ratio?
Are there any areas that were way “off the mark”?
In conclusion, we have looked very briefly at the Golden Ratio and
some of its applications. If you would like more information about
the Golden Ratio, please feel free to visit the following website:
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
The site provides an introduction to the topic, including its
relationship to the Fibonacci sequence, as well as links to many
other websites.
Homework Complete the “Beautiful Faces” worksheet that was
handed out in class. (Teachers: you may access this worksheet at
the Illinois State Board of Education website:
http://www.isbe.net/ils/math/stage_I/7A_7B_7C_8BI.pdf