# How many triangles?

How many triangles?

How many triangles are there in this picture? 94.7% of Americans miss the answer. Try it first, then see the answer below.

6 small red triangles

3 small white triangles

3 medium red triangles with a small white triangle center

1 large triangle with 3 small triangles in the center

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total = 13

### 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/`