Counting of Figures Questions with Solutions

Counting of figures questions ask you to count how many triangles, squares, rectangles or straight lines a complex figure contains. They look simple but are easy to get wrong because shapes overlap and combine. The most reliable approach is systematic counting. Label every point where lines meet, then list the shapes by size: first the smallest single shapes, then shapes made of two parts, then three, and so on. Formulas save a lot of time for standard figures. A triangle with n lines drawn from its top vertex to the base contains n(n + 1)/2 triangles if the n lines include the two sides. A grid of m by n squares contains m(m + 1)n(n + 1)/4 rectangles. Never try to count by eye alone, because it is easy to miss shapes formed by combining smaller ones. Practise the counting of figures questions below. Each solution shows a labelled figure and the count step by step.