Non Verbal Reasoning · Figures & Shapes
Counting Figures — Questions & Answers
Counting figure questions ask how many triangles, squares or rectangles a figure contains, or how many straight lines draw it. Count systematically by size so nothing is missed or counted twice.
- 5 solved questions
- Step-by-step AI walkthroughs
- How-to-approach tips
How to approach
Count by size: the smallest shapes first, then shapes made of 2, 3, 4… pieces, then the whole figure.
Triangle with n lines from the top corner and h lines across: triangles = n(n+1)/2 × (h + 1), counting its sides among the n lines.
Grid of r × c squares: rectangles = C(r + 1, 2) × C(c + 1, 2).
5 solved Counting Figures questions
Count every triangle in the figure, large and small. How many triangles are there?
Solution — answer B
Counting triangles by size, smallest first: 4 + 4 + 4 gives 12.
4 steps · reveal one at a timeStep 1: Count smallest triangles
The first step in counting all triangles is to identify and count the smallest ones. According to the solution, there are 4 triangles of this particular size.
Step 2: Count medium triangles
Next, we look for triangles that are a medium size, larger than the smallest ones. The solution indicates that there are another 4 triangles of this size in the figure.
Step 3: Count largest triangles
Finally, we identify and count the largest triangles present in the figure. The solution states there are 4 triangles of this largest size.
Step 4: The answer
So the answer is (B) 12. Summing up all the triangles counted by size, we have 4+4+4=12 total triangles.
Count every square in the figure, large and small. How many squares are there?
Solution — answer C
Counting squares by size, smallest first: 4 + 4 + 1 + 1 gives 10.
5 steps · reveal one at a timeStep 1: Count smallest squares
Begin by identifying and counting all the squares of the smallest possible size within the figure. According to the solution, there are 4 such squares.
Step 2: Count next size squares
Next, look for squares that are a bit larger than the smallest ones. The solution indicates that there are another 4 squares of this particular size.
Step 3: Find larger squares
Continue counting squares by moving to the next larger size. The solution states that there is 1 square of this specific size.
Step 4: Count largest squares
Finally, identify and count the squares of the largest possible size present in the figure. The solution shows there is 1 square of this size.
Step 5: The answer
So the answer is (C) 10. Summing up all the squares counted by their respective sizes gives a total of 4+4+1+1=10 squares.
Count every triangle in the figure, large and small. How many triangles are there?
Solution — answer C
Counting triangles by size, smallest first: 4 + 3 + 2 + 5 + 3 + 4 + 2 + 1 + 3 + 2 + 1 gives 30.
4 steps · reveal one at a timeStep 1: Understand the Goal
The task is to count every triangle present in the given figure, including both large and small ones. A systematic approach is key to avoid missing any.
Step 2: Adopt a Strategy
The most effective way to count all triangles is to categorize them. We will count them by size, starting with the smallest triangles and progressively moving to larger combinations.
Step 3: Perform the Count
Following this method, we identify groups of triangles of various sizes. Summing these groups gives us: 4+3+2+5+3+4+2+1+3+2+1. Adding these numbers together yields a total of 30 triangles.
Step 4: The answer
So the answer is (C) 30. By systematically counting all triangles, grouped by their size from smallest to largest, we arrive at a total of 30.
What is the smallest number of straight lines that can be used to draw this figure?
Solution — answer B
Joining the pieces that lie on one straight line, the figure is made of 8 straight lines; none of them can be merged further.
4 steps · reveal one at a timeStep 1: Understand the Goal
The question asks us to find the minimum number of straight lines required to form the given figure. This means we should look for ways to combine segments.
Step 2: Apply the Strategy
To find the smallest number of lines, we should identify all segments that lie on the same straight path and consider them as a single line. This is called merging collinear pieces.
Step 3: Count the Lines
After carefully joining all pieces that lie on one continuous straight line, we find that the figure is composed of exactly 8 distinct straight lines. No further merging is possible.
Step 4: The answer
So the answer is (B) 8, because, by merging all collinear segments, the figure is formed by exactly 8 straight lines that cannot be reduced further.
Count every triangle and every square in the figure, large and small. How many of each are there?
Solution — answer B
Triangles by size, smallest first: 16 + 16 + 6 = 38. Squares by size, smallest first: 3 + 4 = 7.
3 steps · reveal one at a timeStep 1: Count Triangles
The solution counts triangles by their size, starting with the smallest. It identifies 16 triangles of one size, another 16 of a different size, and 6 of a third size. Summing these gives a total of 16+16+6=38 triangles.
Step 2: Count Squares
Next, the solution counts squares by their size, also starting with the smallest. It finds 3 squares of one size and 4 squares of a larger size. Adding these together results in a total of 3+4=7 squares.
Step 3: The answer
So the answer is (B) 38 triangles, 7 squares. This option correctly matches the total counts of 38 triangles and 7 squares found in the figure.
Free to start
Get exam-ready on Counting Figures
Generate fresh counting figures practice sets in Test Studio, get step-by-step AI explanations the moment you slip, and close the gaps in Socrative Mode.
- Step-by-step AI solutions
- Fresh practice sets on demand
- Guided Socrative Mode drills