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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/
```
Associated Topics:
High School Discrete Mathematics
High School Euclidean/Plane Geometry
High School Geometry
High School Puzzles

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