Counting Figures Practice Questions with Detailed Solutions – Level 3

Counting Figures Level 3 Reasoning Practice: Triangles, Squares, Lines

Counting triangles, squares and rectangles hidden in a figure, and finding the fewest straight lines needed to draw it, are the skills tested in this Level 3 Counting Figures set. Aptitude360 pitches it at the hard end of the topic. A worked solution with each question shows a systematic way to count without missing shapes.

Question types in Counting Figures Level 3

• Counting every triangle in a given figure
• Counting squares in a given figure
• Counting rectangles with squares included in the total
• Finding the minimum number of straight lines needed to draw a figure

Counting formulas and rules to remember

• Triangles from \( n \) lines through one vertex to the base (outer sides included): \( \dfrac{n(n-1)}{2} \)
• Squares in an \( n\times n \) grid: \( \dfrac{n(n+1)(2n+1)}{6} \)
• Rectangles in an \( m\times n \) grid: \( \dfrac{m(m+1)n(n+1)}{4} \)
• Rectangles in an \( n\times n \) grid: \( \left(\dfrac{n(n+1)}{2}\right)^{2} \)

Ways to count figures without errors

• Label every corner and intersection with a letter, then list shapes by their corner letters
• Count by size: single-part shapes first, then shapes made of two parts, then larger ones
• For straight lines, treat a continuous segment as one line even when it crosses several shapes
• Split a crowded figure into simpler pieces, then add shapes that cross the dividing lines

Who this hard set suits

Logical reasoning learners who are comfortable with the easier Counting Figures levels and want tougher figures before competitive exams.