Questions based on Figure Counting Quiz

Challenge your visual reasoning skills with these figure counting questions that test your ability to analyze and count figures within complex geometric designs. Sharpen your attention to detail and learn helpful techniques for figure counting in diagrams and patterns.

1. Counting Rectangles in a Grid

   Given a 2x3 rectangle grid, how many rectangles (of any size) can be found in the grid?

   1. 9
   2. 15
   3. 18
   4. 12

   Explanation: To count rectangles in a 2x3 grid, multiply the number of ways to choose two horizontal lines out of three, and two vertical lines out of four. This is (3 choose 2) × (4 choose 2) = 3 × 6 = 18. The options 12 and 9 result from miscounting either rows or columns. Option 15 may come from forgetting larger rectangles. Only 18 accounts for all possibilities.

2. Identifying Triangles in an Overlapping Figure

   If two equilateral triangles of the same size overlap completely, how many distinct smaller triangles are formed within the overlapped figure?

   1. 6
   2. 3
   3. 5
   4. 4

   Explanation: When two equilateral triangles overlap perfectly, they create six smaller triangles: three at each intersection plus the central overlap. The option 4 comes from forgetting overlap zones, while 3 misses the central area. Option 5 ignores one of the intersections. The correct count is 6.

3. Squares Within a Composite Shape

   Consider an L-shaped figure formed by removing one 1x1 square from a 2x2 square grid. How many squares (of any size) are present in the resulting figure?

   1. 2
   2. 3
   3. 5
   4. 4

   Explanation: The L-shape contains three 1x1 squares and no 2x2 square (since it's incomplete). Answer 4 may come from mistakenly including the missing square. Option 2 counts too few, and 5 incorrectly includes larger nonexistent squares. Only 3 accounts for the remaining 1x1 squares.

4. Counting Parallelograms in a Given Figure

   In a grid made up of three parallel horizontal lines intersected by two diagonal lines, how many parallelograms are formed within the figure?

   1. 8
   2. 6
   3. 4
   4. 2

   Explanation: Each parallelogram is formed by choosing a pair of horizontal lines and a pair of diagonals. There are 3 ways to pick horizontal pairs and 2 ways for diagonal pairs, so 3 × 2 = 6. Options 2 and 4 underestimate the possible combinations, while 8 comes from an overcount. Thus, 6 is the correct total.

5. Hidden Figures in Complex Design

   A hexagon is divided by three lines connecting opposite vertices. How many individual triangles are formed inside the hexagon?

   1. 12
   2. 8
   3. 9
   4. 6

   Explanation: Drawing three lines connecting opposite vertices of a hexagon divides it into exactly 6 equal triangles. Option 9 or 12 may result from double-counting or adding external areas, and 8 comes from assuming extra intersections. Only the 6 triangles formed inside the hexagon are correct in this scenario.