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Triangles within a Triangle


Date: 11/10/96 at 07:38:57
From: Cato H. Jensen
Subject: Geometry, Math Project (GCSE) 1996

This is an English project called "Triangles in Triangles" for I and H 
level. We are doing this in 10th grade.

The question is:  Investigate equilateral triangles of other sizes in 
order to examine the relationship between the size of a triangle and 
the total number of triangles it contains.

The numbers (where n = total number of triangles and b = number of 
triangles in the base) given are :

     b    n
     1    1
     2    5
     3   13
     4   27
     5   48
     6   78
     7  118
     8  170

I need to find a formula for the total number of triangles when there 
are X base triangles.

I have done the rest of the project, but this part has gotten me 
really stuck.


Date: 01/04/97 at 15:58:04
From: Doctor Chaos
Subject: Re: Geometry, Math Project (GCSE) 1996

I think the best way to approach your problem is to examine the 
pattern you were given in order to calculate the total number of 
triangles.

Look at these pictures:

This triangle has 3 little triangles in the base so b=3.

     /\       In this triangle, there are nine (9) little
    /__\      triangles or b^2.  Then there are three (3)
   /\  /\     triangles which are larger (containing 4 little ones)
  /__\/__\    and one (1) largest (containing all 9).
 /\  /\  /\
/__\/__\/__\  9+3+1 = 13 triangles.


This triangle has 4 little triangles in the base so b = 4.

        /\      The total of little triangles is b^2 or 16.
       /__\     The number of bigger triangles (containing 4 little 
      /\  /\    ones) is 7.  The number of still bigger triangles 
     /__\/__\   (containing 9 little ones) is 3.  The number of 
    /\  /\  /\  largest triangles contains all 16 and is 1.
   /__\/__\/__\
  /\  /\  /\  /\  16+7+3+1 = 27 triangles
 /__\/__\/__\/__\

If we examine the numbers of triangles at every level, we may see a 
pattern which may then suggest a formula.

     b = Number of triangles in the base
     l = Number of little triangles
     f = Next largest triangle (contains 4)
     n = Next largest triangle (contains 9)
     s = Next largest triangle (contains 16)
     T = Total number of triangles

     b   l  f  n  s    T  
     1   1  0  0  0    1
     2   4  1  0  0    5
     3   9  3  1  0   13
     4  16  7  3  1   27 
     5  25  ?  ?  ?   48
     6  36  ?  ?  ?   78
     7  49  ?  ?  ?  118
     8  64  ?  ?  ?  170

Organize the data you have already collected and see if there is a 
formula that will help you add up the various sizes of triangles to 
arrive at the total.  You might even need 2 different formulas; one 
for an odd number on the base and one for even numbers.  You never 
know until you try.

Good luck.

-Doctor Chaos,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
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