Triangle Trouble

Let's take a visual approach to this problem. Consider what we would get if we stuck an identical, flipped triangle onto a side of the original triangle:

This creates a parallelogram made up of smaller parallelograms of side length $1cm$ . Clearly there are $3^2=9$ small parallelograms in this example. Each parallelogram is formed by two equilateral triangles of side length $1cm$ , so there are $2*3^2$ small triangles in the diagram. Since the parallelogram is made up of two identical triangles, there are $\frac{2*3^2}{2}=3^2$ smaller triangles in the example triangle. Thus, the number of smaller triangles in a larger triangle of side length $n cm$ , where $n\in\mathbb{Z}$ , is simply $n^2$ . This is perhaps more obvious when we 'skew' the parallelogram horizontally to form a square:

There is also a numerical way of solving this problem. Consider the result of counting the number of smaller triangles in each row of the larger example triangle:

Since each row adds two smaller triangles from the previous row, and the first row has 1 such triangle, we can express the number of triangles in the $n^{th}$ row as $2n-1$ . Recall the formula for the the $n^{th}$ triangle number: $\frac{1}{2}n(n+1)$ . We can use this formula to work out the total small triangles in a larger triangle of side length $ncm$ . Since the number of triangles in the $n^{th}$ row is $2n-1$ , we can include the $-1$ part in the triangle number formula by subtracting $n$ ( $n$ lots of $-1=-n$ ), which gives $\frac{1}{2}n(n+1)-n$ . To deal with the $2n$ part, remember that the triangle number works for a triangle with $n$ items in the $n^{th}$ row, so we simply multiply the part of the formula concerned with $n$ by $2$ , giving $n(n+1)-n$ as a formula for the number of equilateral triangles of side length $1cm$ that can fit into a larger triangle of side length $ncm$ . Multiplying out the bracket gives: $n^2+n-n=n^2$ . Therefore the answer to the question is $\large 125^2=\color{#20A900}{\boxed{15625}}$ .

$\blacksquare$

The area of an equilateral triangle is proportional to the square of its side length. This means the area of an equilateral triangle with side length 1.25 m, or 125 cm, is $125^2 = 15625$ times larger than the area of an equilateral with side length 1 cm. Thus, the large triangle contains 15625 of the small triangles.