Take a circle. Circumscribe a square around the circle and inscribe a square in the circle. Orient both the squares the same way. Now construct a square exactly in between the two squares.

It was believed that this "middle" square had the same area as the circle, and that its perimeter was equal to the circumference of the circle. Is this true? Be sure to explain your answer thoroughly, and mention anything interesting that you notice along the way.

This turned out to be a harder problem than I thought, but I think it's a good one. 56 people got it right, and 72 got it wrong.

Many of those who got it wrong didn't get credit even though their conclusion was right. The reason I didn't give them credit is that they physically measured the circle and the squares and used those figures to calculate the areas and perimeters. The areas and perimeters come out very close, and you can't be sure that the difference isn't just because you didn't measure them accurately. It's a better idea to assign a length to something in the figure, like the radius of the circle. You can say it's 1, or 2, or even r. Then figure out the other lengths based on that figure.

One way to figure out those other lengths is to use the Pythagorean theorem. Julia Le of Minnechaug Regional High School provides a good example of this below, using 10 as the diameter of the circle. And knowing the Pythagorean theorem helps a lot in this problem, though it is not obvious that you should use it.

You can also use r as the radius of the circle and find a more general answer. Both Jennifer Liang of Odle Middle School and Garrett Kluz-Wisnewski of Highland Park Senior High School did it that way. They both included some nice observations - the perimeter/circumference and the areas can't ever be equal at the same time! Read their solutions below.

A couple of people said that it can't work because they had read that you can't "square the circle" - that is, you can't construct a square that has the same area or perimeter as a given circle. That's true, but it doesn't really explore what we have set up in the problem, so I wanted a little more of an explanation.

In a problem like this people can use all different numbers and come up with all different figures for area and perimeter and circumference. So how do you correct things without doing everything for every submission? You do just what Jason Chiu of Laramie Junior High School and Katie Schill of Shaler Area High School did - you find the "percent error" in the answers. How many percent off is the area? How about the perimeter/circumference? I would check anyone's answers by finding the percent errors between their numbers and see if they matched what I knew were the right figures. No matter what numbers you use, if you're right, the percent error will always be the same. You can read Katie's solution below.

One thing to note is that you can't average areas - so you can't find the areas of the big and little squares and average them to find the area of the middle square. But you can average perimeters... why is that? (Maybe that would make a good PoW!)

Rick Peterson presented some good ideas about how you could go back in time and try to convince people what pi was really equal to. I haven't included his solution, but you should go read it in the full solutions.

When you send in your solution, please proofread it before you send it, and make sure you include your full name. Especially proofread your email address, because if you get it wrong, I can't always figure it out, so you might not get a response!

A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.

From:   Julia Le
        
Grade:  11
School: Minnechaug Regional High School, Wilbraham, Massachusetts

No, it is not true.  Here's the explanation:

Let's pretend that circumscribed square has a side of 10.  Therefore the 
diameter of the circle is equal to 10.  That means that the radius of the circle 
is equal to 5.  Now, let's daw a line from the center of a circle  to two 
adjacent corners of the inscribed square.  This forms a 45-45-90 triangle.  The 
two drawn lines are each equal to 5 because they are both radii of the circle.  
Using my knowledge of 45-45-90 triangles, the length of a side of the inscribed 
square is 5*2^(1/2).  (Five square roots of two.)  Add this number to 10 and 
divide by 2.  This gives you the length of a side of the middle square.  The 
length is 8.53.  From this the perimeter of the middle square is 34.14.  This 
does not correspond to the circumference of the circle which is 31.4.  
(circumference=pi*diameter).  The areas of the middle square and the circle do 
no correspond either.  The area of the square is 72.86.  The area of the circle 
is 78.54.  (area=pi*radius^2).  

From:   Jennifer Liang
        
Grade:  8
School: Odle Middle School, Bellevue, Washington

Teacher: Art Mabbott

Dear Annie,
     This is my answer for this week's POW:

radius of circle= r
center of circle= O
vertex of outer square= F
vertex of middle square= B and A
   so AB is the side of the middle square
vertex of inner square= E

(OF)^2= r^2 + r^2

OF=sqrt(2)r
BF= 1/2(OF - OE)= 1/2(sqrt(2)r - r)
OB= OF - BF= sqrt(2)r - 1/2(sqrt(2)r - r)= 1/2(sqrt(2)r + r)
(AB)^2= (OA)^2 + (OB)^2= 2(OB)^2= 2[1/2(sqrt(2)r + r)]^2
AB= sqrt(2)[1/2(sqrt(2)r + r)]= 1/2(2r + sqrt(2)r)

for middle square:		
  area= (AB)^2= 2[1/2(sqrt(2)r + r)]^2 is about 2.914r^2
  perimeter= 4AB= 4r + sqrt(2)2r is about 6.828r

today pi is about 3.1416
for the circle:
  area= (pi)r^2 is about 3.1416r^2
  circumference= 2(pi)r is about 6.283r

  When we compare the area and the perimeter/circumference, we discover that the 
areas of the middle square and circle is not equal, and neither is the 
perimeter/circumference.

  I noticed that if the circle and square have the same area, the "pi" that was 
used back then was about 2.9.  But when we use their "pi" to see if the 
circumference and perimeter are the same, we discover that they're still not.  
So even if the area is the same, the circumference/perimeter cannot be the same.  
That means that it isn't possible for the area and circumference/perimeter to be 
equal at the same time.

From:   Katie Schill
        
Grade:  10
School: Shaler Area High School, Pittsburgh, Pennsylvania

Subject: pow

Hi!  My name is Katie Schill, and I'm a tenth grader at Shaler 
Area High School.
	The method that was used long ago to approximate the area and 
circumference of a circle by inscribing and circumscribing squares around 
a circle and constructing a square exactly between the first two squares 
is not accurate.
	For example, take a circle with a diameter or 2.  The large 
square's edge would be 2 units in length, and the diagonals of the small 
square are each 2 units.  Since the length of the diagonals of the small 
square are known, it is possible to find the length of each side using 
the Pythagorean Theorum, a^2 + b^2 = c^2 because each diagonal cuts the 
square into two right triangles.  a and b are the lengths of two of the 
sides of the triangle, and c is the length of the hypotenuse, or the 
diagonal of the square.  Because all four sides of a square are equal, a 
= b.  When 2 is squared, divided in half, and the square root is taken, a 
= b = sqrt(2) which is approximately 1.4142.  The length of a side of th 
large square is 2, and the length of a side of the small square is 
1.4142.  When these two numbers are subtracted and th difference is 
divided by two, the number found can be added to the length of a side of 
the small square to find the length of a side of the middle square.  A 
side of the middle square is 1.7071 units long.
	The length of a side of the middle square can be multiplied by 
four to find the perimeter of squared to find the area of the middle 
square.  
the perimeter is 6.8284 units, and the area is 2.9142 square units.  The 
circumference of the circle can be found by multiplying pi by the 
deameter of the circle.  The circumference is 6.2832 units.  the area of 
the circle can be found by multiplying pi times the radius squared.  The 
area of the circle is 3.1416 square units.
	The difference between the circumference of the circle and the 
perimeter of the square is .5452 units.  The error is about 9%.  The 
difference between the area of the circle and the area of the square is 
.2274 units.  The error is about 8%.  Whereas the approximate 
circumference and area of the circle are not very precise, they are not 
far from the accurate information.

From:   Garrett Kluz-Wisnewski
        
Grade:  9
School: Highland Park Senior High School, St. Paul, Minnesota

Garrett Kluz-Wisnewski, Grade 9, Algebra Two IB
Highland Park Senior High School, (612) 293-8940
www.sdtpaul.k12.mn.us/hphs/highland.html
Geometry POW September 22-26, 1997


Part #1
The circumference of the circle cannot be equal to
he perimeter of the middle square.

the radius of the circle = r
the diameter of the circle = 2r
the length of a side of the big square is equal to the diameter
perimeter(big square) = 8r

the length of a side of the little square is equal to r(radical 2)
perimeter(little square) = 4r(radical 2)

the length of a side of the middle square is equal to
the average of the sides of the other two squares, or
(2r + r(radical 2))/2, so
perimeter(middle square) = 2(2r + r(radical 2) = 2r(2 + radical 2)

circumference(circle) = 2r(pi)

THEREFORE if the perimeter(middle square) = circumference(circle),
then (2 + radical 2) must be equal to (pi). But this isn't true, so
the circumference of the circle and the perimeter of the middle square
cannot be equal.

Part #2
The area of the circle cannot be equal to the area of the middle square.

the radius of the circle = r
the diameter of the circle = 2r
the length of a side of the big square is equal to the diameter, so
area(big square) = 4r^2

the length of a side of the little square is equal to r(radical 2)
area(little square) = 2r^2

the length of a side of the middle square is equal to
the average of the sides of the other two squares, or
(2r + r(radical 2))/2 = (r/2)(2 + radical 2)
area(middle square) = (r^2/4)(4 + 4(radical 2) + 2) = (r^2)((6 + 4(radical 
2))/4)

area(circle) = (r^2)(pi)

THEREFORE if the area(middle square) = area(circle),
then (6 + 4(radical 2))/4) must be equal to (pi). But this isn't true, so
the area of the circle and the area of the middle square cannot be equal.

----------

I noticed that the perimeter of the big square is radical(2) times
the perimeter of the small square. Also, the area of the big square
is 2 times the area of the small square.

I also noticed that the perimeter of the middle square is slightly larger
than the perimeter of the circle but that the area of the circle is
slightly larger than the area of the middle square.

The following students submitted correct solutions this week: