Counting Figures

What the question asks

You get a line drawing and must say how many triangles are hidden in it. A triangle is any closed shape with exactly three straight sides. The sides may be made of several small drawn pieces joined in a straight line. Small triangles, medium ones made of several pieces, and the big outer triangle all count.

Think of a samosa cut by a knife. Each cut makes new small pieces, but a big piece made of two small pieces is still a triangle. Most wrong answers come from missing these bigger joined triangles.

Method 1: size by size

1. Count the smallest triangles (the ones with no line inside).
2. Count triangles made of 2 pieces, then 3 pieces, then 4, and so on.
3. Do not forget the whole figure if it is itself a triangle.
4. Add all the groups.

Example: a square with both diagonals. Smallest triangles = 4 (meeting at the centre). Triangles made of 2 pieces = 4 (each is half of the square). Total = 8.

Method 2: name the corners

Put a letter on every corner and every crossing point. Take one point at a time and list every triangle that has it as a corner, using only points joined by drawn lines. This is slow but safe for odd, irregular figures.

Ready formulas for standard figures

Lines from the apex. A triangle with k lines drawn from the top corner (apex) to the base splits the base into k + 1 parts. Every triangle here has the apex as its top and two of the lines through the apex as its sides. There are k + 2 such lines, so

$T = \frac{(k+1)(k+2)}{2}$

With 3 apex lines: $\frac{4 \times 5}{2} = 10$.

Adding horizontal cuts. Each line parallel to the base that crosses all the apex lines gives one more full set. With h such lines: $T = \frac{(k+1)(k+2)}{2} \times (h+1)$. With 2 apex lines and 1 cut: $6 \times 2 = 12$.

Triangle cut into rows of small triangles. A big triangle whose sides are divided into n equal parts and joined by lines parallel to the sides:

Remember that upside-down triangles also count (a 3-row figure has 3 upside-down small ones).

Square with diagonals. Both diagonals: 8. Both diagonals and both midlines: 16 (8 smallest + 4 of two pieces + 4 half-squares).

Triangle with its three medians (lines from each corner to the middle of the opposite side): 16 triangles.

Pentagon with all its diagonals (a star inside a pentagon): 35 triangles. This one appears again and again, so remember it.

Special cases

• A four-sided piece is never a triangle, even if it looks pointed.
• Three points on one straight line do not make a triangle.
• Two drawn pieces that lie in one straight line act as one side.
• When a figure is symmetric, count one half and double, but add any triangle that sits across the middle line only once.

Quick revision

• Count size by size: smallest, then 2-piece, 3-piece, and finally the whole figure.
• Apex lines: (k+1)(k+2)/2; with h horizontal cuts multiply by (h+1).
• Square + 2 diagonals = 8; add 2 midlines = 16.
• Triangle rows: 1, 5, 13, 27, 48.
• Triangle with 3 medians = 16; pentagon with all diagonals = 35.
• Name the corners for any irregular figure.