Counting Figures Practice Questions with Detailed Solutions – Level 2

Counting Figures Level 2 Reasoning Practice: Triangles, Squares and Lines

This Level 2 Counting Figures set is medium-level reasoning practice. It checks whether you can count every triangle, square and rectangle inside a figure, including overlapping shapes, and find the minimum number of straight lines needed to draw it. On Aptitude360, every question has a worked solution that breaks the figure down piece by piece.

What this Counting Figures set asks you to count

• Triangles formed by inner and crossing lines
• Squares of every size within a figure
• Rectangles, with squares included
• Minimum number of straight lines needed to draw a figure

Counting figures formulas for standard patterns

• Squares in an \( n \times n \) grid: \( \dfrac{n(n+1)(2n+1)}{6} \)
• Rectangles in an \( m \times n \) grid: \( \dfrac{m(m+1)}{2}\times\dfrac{n(n+1)}{2} \)
• Triangles when \( n \) lines (both sides included) join the apex to the base: \( \dfrac{n(n-1)}{2} \)

Core ideas for accurate shape counting

• Labelling every corner and intersection point with a letter
• Separating single-piece shapes from those formed by combining pieces
• Treating collinear segments as one straight line

Revision tips for figure counting questions

• Count the smallest shapes first, then shapes made of two, three or more pieces
• Split a complex figure into a known pattern and apply its formula before adding extra shapes
• For straight-line questions, count horizontal, vertical and slanting lines separately

Who this Level 2 set suits

Reasoning aspirants who know the basics of figure counting and want medium-level practice before denser figures.