Counting Figures — Triangles, Squares, Rectangles and Straight Lines: the Formulas That Beat the Eye

The Count Your Eye Always Gets Wrong

Ask ten students how many triangles are in a figure and you will get ten answers. That is exactly why examiners print it. Counting figures is the one reasoning question where confident guessing is punished, because the correct answer is almost always larger than the eye's count.

Two skills settle every question of this type:

• ›Formulas — for regular figures such as grids, polygons and points on a circle, a counting formula gives the exact answer in seconds.
• ›Size-class counting — for irregular composite figures you count by size in order: every one-unit shape first, then two-unit, then three-unit. Never randomly.

The errors are predictable. Students count only the smallest shapes and miss the larger ones built out of them. They count upward triangles and forget the downward ones. They count squares when the question asked for rectangles — and every square is also a rectangle.

Learn the five formulas that follow and this becomes a free mark.

1. How many rectangles are there in a grid of 4 columns and 3 rows of unit squares?

2. How many squares of all sizes are there on a standard 8 × 8 chessboard?

3. How many diagonals does a decagon (a polygon with 10 sides) have?

4. Twelve points lie on a circle, no three of them in a straight line. How many straight lines can be drawn by joining them?

5. Each side of an equilateral triangle is divided into 3 equal parts and lines are drawn parallel to the sides through the division points. How many triangles of all sizes are formed?

6. The base of a triangle is divided into 4 equal parts and the apex is joined to each of the three division points. How many triangles are formed in the figure?

7. On an 8 × 8 chessboard, how many rectangles are there that are NOT squares?

8. There are 10 points in a plane, of which 4 lie on one straight line and no other three are collinear. How many triangles can be formed?