Quinn Cunningham
Grade 9, Germantown Academy, Fort Washington, Pennsylvania

For a six-sided rectangle, to find the side lengths in a circle with a diameter of 15 units or less, you can use the equation: a^2 +b^2 + c^2 = d^2, where "d" equals the diameter of the circle. This is because the diameter of the circle is the diagonal inside the 3-D square.

With this information, I thought of all the possible combinations (with three numbers) using the numbers 1 to 15. Once I had found them all (about four pages written), I created a calculator program so that I could figure out the answers more quickly. Only certain combinations of numbers can be the answer to this problem because the sides all have to be integers and the diameter ("d") must also equal an integer of fifteen or less. There were combinations that were found which equaled a diameter of more than fifteen, but there were fourteen combinations that equaled a diameter of fifteen or less. Not every number from 1 - 15 had a combination that equaled a diameter that was an integer. Those that did though were: (all numbers listed would be squared in the equation)

"a-side"   "b-side"   "c-side"   Diameter of circle
   2	       2	1               3
   4	       4	2	        6
   2	       3	6	        7
   1	       8	4	        9
   4	       4	7	        9
   6	       6	3	        9
   6	       2	9	       11
   6	       6	7	       11
   8	       8	4	       12
   4	       3	12	       13
   6	       4	12	       14
  10	      10	5	       15
  10	       2	11	       15
   5           2	14	       15

Here's some more information about the program: My program (called Roundsqu) first gives a menu for whether you want to find side A, B, C, or the diameter of the circle (or the diagonal within the square). Within each of these options is a manipulation of the equation (A^2 + B^2 + C^2 = D^2) depending upon which side you choose to find. After making a list of all the possible combinations of three numbers from 1-14, I just used the equation to find the diameter and plugged in all the combinations I found. The ones that came up with integers as answers, I wrote down and sent in. All of this was done on a TI-83.

Here's the program:

PROGRAM: ROUNDSQU
:ClrHome
:Lbl  A
:Menu("Choose A Side", "Side A", 1, "Side B", 2, "Side C", 3, "circle Diam", 
4, "Exit",5)

:Lbl  1
:Prompt B, C, D
:(Square root of)* (D^2 - B^2 - C^2) = A   *there is a symble there, not
: Disp  "THE A SIDE IS", A                  the actual words
:Return
:Lbl 2
:Prompt A, C, D
:(Square root of) (D^2 - A^2 - C^2) = B
:Disp  "THE B SIDE IS" , B
:Return
:Lbl 3
:Prompt A, B, D
:(Square root of) (D^2 - A^2- B^2) = C
:Disp  "THE C SIDE IS", C
:Return
:Lbl 4
:Prompt A, B, C
:(Square root of) (A^2 + B^2 + C^2) = D
:Disp  "THE DIAMETER IS", D
:Return
:Lbl  5
:Stop

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