Counting Triangles, Squares and Rectangles in Figures: Systematic Methods

Counting-figure questions show a drawing made of straight lines and ask how many triangles, squares or rectangles it contains. The answer is almost never the number of small shapes you can see. Larger shapes are made by combining smaller ones, and the usual mistake is to miss some of those, or to count one twice.

Two habits fix this. First, label every point where lines meet with a letter, so each figure can be named by its corners. Second, count in a fixed order: by size, smallest first, or by a fixed starting corner. For common figure types there are also formulas, which give the answer in seconds and act as a check on your count.

Take a triangle and draw straight lines from its top vertex down to the base. Every pair of lines from that vertex, together with the base, makes a triangle. So if there are k lines from the vertex in total, counting the two sides, the number of triangles is the number of ways to choose 2 of them: C(k, 2).

If lines are also drawn across the triangle parallel to the base, each one acts as another base. Multiply by the number of such lines, including the base itself.

In an n × n grid of small squares, count squares by size. There are n² squares of side 1, (n − 1)² of side 2, and so on down to 1 square of side n. The total is the sum of the first n square numbers:

For a rectangular grid m squares tall and n wide, with m smaller, count each size separately: squares of side k number (m − k + 1)(n − k + 1), for k from 1 to m.

Every rectangle in a grid is fixed by choosing two horizontal lines and two vertical lines. A grid m squares tall and n wide has m + 1 horizontal lines and n + 1 vertical lines.

A square with both diagonals drawn is a common building block. The diagonals meet at the centre and cut the square into 4 small triangles. Each half of the square cut off by one diagonal is a larger triangle, and there are 4 such halves.

An equilateral triangle divided into small equilateral triangles, with n along each side, contains triangles pointing up and pointing down. Count both. For small n it is quick to count by size; the total also follows a formula, n(n + 2)(2n + 1) ÷ 8, rounded down.

• ✘ Counting only the smallest shapes and missing the combined ones.
• ✘ Forgetting the two sides of the triangle when counting lines from a vertex.
• ✘ Missing the pointing-down triangles in a triangular grid.
• ✘ Using the n × n square formula on a grid that is not square.

Other spatial topics are dice problems and cube cutting and painting. More reasoning methods are on the study hub.

What is the best general method when no formula fits?

Label every point where lines meet. Then count by size or by a fixed starting point, listing each figure by its corner letters. Listing by letters makes it easy to spot a figure counted twice or one you missed.

Does the rectangle count include squares?

Yes. A square is a rectangle, so the C(lines, 2) × C(lines, 2) formula counts squares too. If a question asks for rectangles that are not squares, subtract the square count from the rectangle count.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.