Mathematical Investigation: The Perfect Rectangle
Topic: Angles, Polygons and Geometric Constructions
The purpose of this worksheet is to investigate what is so nice about perfect rectangles.
Section A: The Perfect Rectangle
1.
Which one of the following rectangles do you like best? It is o.k. if your opinion is different
from your classmates or friends. So no mark for this question.
B
A
F
D
C
E
Your choice: _________
2.
The diagram below shows a series of squares. The two smallest squares S1 and S2 have
lengths of 1 unit. Complete the middle column of the table below.
[1]
S7
S8
S3
S2 S1
S6
S4
\4
S5
Square Sn
Length of Square Ln
Ln / Ln–1 (to 8 s.f.)
S1
1
–
S2
1
1/1=1
S3
S4
S5
S6
S7
S8
© Joseph Yeo
1
3.
What do you notice about the numbers in the middle column of the above table?
4.
Complete the last column of the above table, leaving the answer to 8 significant figures
where applicable.
[1]
5.
What is the significance of the ratio L8 / L7?
[1]
Hint: Ratio of something for some rectangle. Use a pencil to highlight the rectangle in Q2. [1]
6.
Is the rectangle in Q5 about the same shape as Rectangle C in Q1?
7.
Is your favourite rectangle Rectangle C in Q1? It is o.k. if your choice is different. [no mark]
8.
Rectangle C is called the Perfect Rectangle because it is supposed to be the most pleasing to
the eye for most people. Measure the length and the width of Rectangle C and calculate the
ratio of its length to its width.
[1]
9.
The ratio Ln / Ln–1 in the last column of the above table converges to the value 1.61803 39887
49894 84820 45868 34365… This value is called the Golden Ratio  (pronounced as ‘phi’).
Is this value about the same as your answer in Q8? What does this mean?
[1]
10.
Use a calculator to evaluate
1 5
and write down its value. What do you notice?
2
[1]
[1]
[1]
Section B: Some Interesting Properties of the Golden Ratio
11.
Use a calculator to evaluate 2 and write down its value, correct to 5 decimal places. What do
you notice? How is 2 related to  (other than squaring )?
[1]
12.
Use a calculator to evaluate 1/ and write down its value, correct to 5 decimal places. What
do you notice? How is 1/ related to  (other than taking the reciprocal of )?
[1]
© Joseph Yeo
2
Section C: Perfect Rectangle and Golden Ratio in Real Life
13.
Many buildings and structures are shaped like the Perfect Rectangle or linked to the Golden
Ratio. For example, the diagrams below show a photo of the Parthenon at Athens. Measure
the length and width of the rectangle in the first diagram below (must take into account the
height of the original roof) and calculate the ratio of its length to its width. What type of
rectangle do you get?
[2]
14.
Another example is the Great Pyramid built by the Egyptians in Giza
(see below). Calculate the ratio of the “distance up the middle of one
side” to the “distance from the edge to the centre”. What do you get? [2]
Distance up middle of
one side = 186.4 m
Distance from edge
to centre = 115.2 m
© Joseph Yeo
3
15.
Nature also has some interesting shapes. A nautilus is a creature that lives in the sea and it
grows to about 25 cm (see photos below). As its shell develops, it creates one of the most
fascinating shapes in mathematics. To see how the shell looks like, an arc of a quadrant is
drawn with its centre at the bottom left hand corner of S8 (see diagram below).
S7
S8
S3
S2 S1
S6
S4
\4
S5
(a)
(b)
(c)
(d)
© Joseph Yeo
Draw a similar arc for S7 but with the centre at the bottom right hand corner of S7.
Draw a similar arc for S6 but with the centre at the top right hand corner of S6.
Draw a similar arc for S5 such that the arcs form a curve that spirals inwards.
Continue the process until you reach S1.
[2]
4
Further Investigation
16.
One of the most important questions in any mathematical investigation that you should ask
yourself is: “What else is there for me to investigate?” List one thing related to this topic that
you would like to investigate further and if time permits, investigate it.
[2]
Question: __________________________________________________________________
Answer:
Conclusion
17.
Write down one main lesson that you have learnt from this worksheet.
Final Score:
/ 20
Final Score
16-20
13-15
10-12
7-9
0-6
Grade
A
B
C
D
F
Teacher’s Comments (if any):
© Joseph Yeo
[1]
5
Mathematical Investigation: The Perfect Rectangle
Topic: Angles, Polygons and Geometric Constructions
Answers & Scoring Rubric
(Marks allocated in square brackets; total marks = 20)
Section A: The Perfect Rectangle
1.
No mark for this question because it is a matter of personal preference. The teacher is to
collate the results for the class and observe whether most students choose Rectangle C. But
don’t tell them the significance of C (Perfect Rectangle – see later) but can tell them the
significance of B (shape of A4 paper) and F (square – a special rectangle). It does not matter
if the majority does not prefer Rectangle C because they may be so used to A4 paper
nowadays that they may prefer Rectangle B. But one of the authors has surveyed some
students recently and most of them still prefer Rectangle C.
Q2.
[1 mark for middle column]
Q4.
[1 mark for last column]
Square Sn
Length of Square Ln
Ln / Ln–1 (to 8 s.f.)
S1
1
–
S2
1
1/1=1
S3
2
2/1=2
S4
3
3 / 2 = 1.5
S5
5
5 / 3 = 1.6666667
S6
8
8 / 5 = 1.6
S7
13
13 / 8 = 1.625
S8
21
21 / 13 = 1.6153846
3.
Fibonacci numbers or sequence
[1]
5.
The rectangle is highlighted below: its length is L8 and its width is L7.
Therefore the ratio L8 / L7 is the ratio of the length to the width of this rectangle.
L7
S7
S8
L8
S3
S2 S1
S6
S4
\4
S5
© Joseph Yeo
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[1]
[1]
6.
Yes
[1]
7.
See Comments for Q1 above. Teacher can now tell the students the significance of Rectangle
C and compare it with their preference. [No mark for this question]
8.
2.8 / 1.75 = 1.6
9.
Yes. Ratio of length of Perfect Rectangle to its width is the Golden Ratio.
10.
1 5
is the exact value of the Golden Ratio. [1]
2
[1]
[1]
Section B: Some Interesting Properties of the Golden Ratio
11.
2 = 2.61803 (to 5 decimal places) =  + 1
[1]
2
1  5 
  6  2 5  3  5    1]
[For your info:  2  

4
2
 2 
12.
1/ = 0.61803 (to 5 decimal places) =  – 1
[For your info:
[1]


1
2
2
1 5 21 5
5 1





  1]
 1 5 1 5 1 5
1 5
2
Section C: Perfect Rectangle and Golden Ratio in Real Life
13.
5.6 / 3.5 = 1.6  
Perfect Rectangle
[1]
[1]
14.
186.4 / 115.2 = 1.6180 5  
Golden Ratio
[1]
[1]
15.
See diagram below. OHT template is on page 9.
[2 marks for correct construction; 1 mark for incomplete construction or minor mistake]
S7
S8
S3
S2 S1
S6
S4
\4
S5
© Joseph Yeo
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Further Investigation
16.
Some suggestions for further investigations: [2] {Full marks for creative or original idea}
 How do you find the limit of convergence of the ratio Ln / Ln–1?
Ans: One way is to use an Excel spreadsheet. The gist of it is: B4=1, B5=1, B6=B4+B5,
and then auto-complete for B7 onwards. For more info, download Fibonacci.xls
from http://math.nie.edu.sg/bwjyeo/aa.
 How do you construct a perfect rectangle with straightedge and compasses?
Ans: See below or Rectangle.gsp (download from http://math.nie.edu.sg/bwjyeo/aa). The
gist of it is: ABCD is a square with AB = 1 unit; E is the midpoint of AD so that
5
1
5
1 5
EC =
and AF = AE + EF = 
=
.
2
2
2
2
F
D
C
E
A
B
Conclusion
17.
Some suggestions for main lesson learnt from this worksheet:
[1]
 Perfect Rectangle is the most pleasing to the eye.
 Perfect Rectangle and Golden Ratio occur in man-made structures and in nature.
 Mathematics occurs in nature.
© Joseph Yeo
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Shell of Nautilus
S7
S8
S3 S4
S2 S1
S6
S5
© Joseph Yeo
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