Counting of Figures: Types, Methods and Examples
Counting of figures is a non-verbal reasoning topic in which a geometrical drawing made of lines is given, and the candidate must count the total number of a particular figure, such as triangles, squares or rectangles, that the drawing contains. The figures may be of any size, so the candidate must count the small figures, the medium figures formed by joining them, and the largest figure as well. The drawing usually looks simple, but the number of hidden figures is often much larger than it appears at first sight.
This topic is asked in APSC, ADRE, Assam Police, DHS, DME, SSC and Railway examinations because it tests observation, patience and systematic counting. It rewards a candidate who counts in an organised way instead of guessing. This article gives a complete deep-study revision of counting of figures. It explains the basic terms, the standard shapes and their counting methods, the types of questions, the step-by-step approach, the common mistakes and a practical preparation plan.
Basic Terms
- Figure: the geometrical drawing given in the question.
- Triangle: a polygon with three sides and three angles.
- Square: a quadrilateral with four equal sides and four right angles.
- Rectangle: a quadrilateral with four right angles.
- Base line: a line from which the other lines are drawn.
- Small figure: the smallest indivisible figure in the drawing.
- Composite figure: a larger figure formed by joining smaller ones.
- Diagonal: a line joining two opposite corners of a figure.
- Region: a part of the drawing enclosed by lines.
- Systematic counting: counting figures in a fixed order so that none is missed or repeated.
Core Concepts and Standard Methods
- A figure of any size counts, as long as it is formed by the lines of the drawing.
- Count the smallest figures first, then the figures formed by two parts, then three parts, and so on.
- In a triangle divided by lines drawn from one vertex, the number of triangles is found by counting the regions and their combinations.
- In a square grid of n rows and n columns, the number of squares is the sum of the squares of the numbers from n down to one.
- In a grid of m rows and n columns, the number of rectangles is found by multiplying the number of ways of choosing two rows and two columns.
- For triangles formed by intersecting lines, mark each region with a label and combine the regions in order.
- Symmetry in the drawing can be used to reduce the work of counting.
- Always verify the total by counting in two different orders.
Types of Counting of Figures Questions
The first type asks for the number of triangles in a drawing. The second type asks for the number of squares or rectangles. The third type asks for the number of circles or quadrilaterals, or a mix of shapes. The fourth type gives a large square or triangle divided into many parts and asks for a specific figure. A fifth type gives a straight line with several points on it and asks for the number of triangles or segments formed. In every type the answer is found by counting systematically rather than by a single formula, though the standard formulas are useful for regular grids.
Steps to Count Figures
First, study the drawing and mark the smallest regions with numbers or letters. Second, count the smallest figures of the required shape. Third, count the figures formed by combining two, three and more regions, taking care not to repeat a figure already counted. Fourth, count the largest figure if it is of the required shape. Fifth, add all the counts to get the total. Finally, recheck the total by counting in a different order or by using the standard formula for a regular grid.
Examination Trends
In competitive examinations counting of figures usually carries three to five marks. Triangles and squares are the most commonly asked shapes, and the drawing is usually a square or a triangle divided by lines. The number of figures often follows a neat pattern, such as a sum of squares, so a candidate who knows the standard results can answer quickly. Because the topic is purely visual and needs no calculation beyond simple addition, careful counting is the only requirement.
Common Mistakes to Avoid
- Do not count only the small figures and forget the composite ones.
- Do not count the same figure twice in two different combinations.
- Do not assume a shape is a square when it is only a rectangle.
- Do not ignore the largest outer figure.
- Do not guess the total without a systematic order.
- Do not rush; counting errors are the main cause of lost marks here.
How to Prepare This Topic
Begin by learning the standard results for a triangle divided by lines from one vertex and for a square grid of two, three and four rows. Then practise counting triangles and squares in simple drawings, always writing the count for each size separately before adding them. After that, move to drawings that combine a square and its diagonals, and to figures with overlapping triangles. Keep a small notebook of the standard formulas and of the drawings whose answers surprised you. Daily practice of ten to fifteen figures for a few days makes the counting both fast and accurate, and it trains the eye to see hidden figures that an untrained candidate would miss.
Another useful habit is to label every region of the drawing with a number before counting, and then to list the combinations in words, such as one-two, two-three and one-two-three. This written method prevents double counting and makes the total easy to verify. When the drawing is symmetrical, count one half and multiply, taking care about the figures that cross the line of symmetry. Remember that the answer must account for figures of every size, from the smallest to the outermost. With systematic practice, counting of figures becomes a dependable source of marks in every competitive examination.
Frequently Asked Questions
What is counting of figures in reasoning?
It is a non-verbal reasoning topic in which a geometrical drawing is given and the candidate must count the total number of a particular figure, such as triangles or squares, including figures of every size.
How do I count triangles without missing any?
Label the smallest regions, count the small triangles, then the triangles formed by two, three and more regions, and finally the largest triangle, adding the counts in order.
How many squares are there in a square grid?
In a square grid of n rows and n columns, the number of squares is the sum of the squares of the numbers from n down to one.
Why do I lose marks in counting figures?
Most marks are lost by counting the same figure twice or by forgetting the composite figures. Counting in a fixed order and rechecking in a second order avoids this.
Which exams ask counting of figures questions?
APSC, ADRE, Assam Police, DHS, DME, SSC and Railway examinations ask counting of figures questions in the non-verbal reasoning section.