Sure! Here are a few examples of counting figures with answers:
Example 1: Counting Shapes
Question: How many triangles are there in the figure below?
/\
/ \
/____\
/\ /\
/ \ / \
/____\/____\
Answer:
- Small triangles: 8
- Medium triangles (composed of 2 small triangles each): 3
- Large triangles (composed of 4 small triangles each): 1
Total number of triangles: 8 + 3 + 1 = 12
Example 2: Counting Squares
Question: How many squares are there in a 3×3 grid?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- 1×1 squares: 9
- 2×2 squares: 4
- 3×3 squares: 1
Total number of squares: 9 + 4 + 1 = 14
Example 3: Counting Rectangles
Question: How many rectangles are there in a 2×2 grid?
_ _
|_|_|
|_|_|
Answer:
- 1×1 rectangles: 4
- 1×2 rectangles: 4
- 2×1 rectangles: 4
- 2×2 rectangles: 1
Total number of rectangles: 4 + 4 + 4 + 1 = 13
Example 4: Counting Figures in a Complex Shape
Question: How many rectangles and squares are there in the figure below?
_______
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 16
- 2×2 squares: 9
- 3×3 squares: 4
- 4×4 square: 1
Total number of squares: 16 + 9 + 4 + 1 = 30
- Rectangles (including squares):
- 1×1 rectangles: 16
- 1×2 rectangles: 12
- 1×3 rectangles: 8
- 1×4 rectangles: 4
- 2×1 rectangles: 12
- 2×2 rectangles: 9
- 2×3 rectangles: 6
- 2×4 rectangles: 3
- 3×1 rectangles: 8
- 3×2 rectangles: 6
- 3×3 rectangles: 4
- 3×4 rectangles: 2
- 4×1 rectangles: 4
- 4×2 rectangles: 3
- 4×3 rectangles: 2
- 4×4 rectangle: 1
Total number of rectangles (including squares): 16 + 12 + 8 + 4 + 12 + 9 + 6 + 3 + 8 + 6 + 4 + 2 + 4 + 3 + 2 + 1 = 100
Example 5: Counting Triangles in a Star
Question: How many triangles are there in the star figure below?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 10
- Medium triangles (composed of 2 small triangles each): 5
- Large triangles (composed of 4 small triangles each): 1
Total number of triangles: 10 + 5 + 1 = 16
Example 6: Counting Figures in a Complex Grid
Question: How many triangles and rectangles are there in the figure below?
____
/\ /
/__\/
/\ /
/__\/
\ /\
\/__\
Answer:
- Triangles:
- Small triangles: 5
- Medium triangles: 2
- Large triangles: 1
Total number of triangles: 5 + 2 + 1 = 8
- Rectangles (including squares):
- 1×1 rectangles: 3
- 1×2 rectangles: 2
- 2×1 rectangles: 2
- 2×2 rectangles: 1
Total number of rectangles: 3 + 2 + 2 + 1 = 8
Example 7: Counting Shapes in a 4×4 Grid
Question: How many squares and rectangles are there in a 4×4 grid?
_ _ _ _
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 16
- 2×2 squares: 9
- 3×3 squares: 4
- 4×4 square: 1
Total number of squares: 16 + 9 + 4 + 1 = 30
- Rectangles (including squares):
- 1×1 rectangles: 16
- 1×2 rectangles: 12
- 1×3 rectangles: 8
- 1×4 rectangles: 4
- 2×1 rectangles: 12
- 2×2 rectangles: 9
- 2×3 rectangles: 6
- 2×4 rectangles: 3
- 3×1 rectangles: 8
- 3×2 rectangles: 6
- 3×3 rectangles: 4
- 3×4 rectangles: 2
- 4×1 rectangles: 4
- 4×2 rectangles: 3
- 4×3 rectangles: 2
- 4×4 rectangle: 1
Total number of rectangles (including squares): 16 + 12 + 8 + 4 + 12 + 9 + 6 + 3 + 8 + 6 + 4 + 2 + 4 + 3 + 2 + 1 = 100
Example 8: Counting Figures in an Overlapping Shape
Question: How many distinct geometric shapes (triangles, squares, and rectangles) are there in the figure below?
____
/\ /
/__\/
/\ /
/__\/
\ /\
\/__\
Answer:
- Triangles:
- Small triangles: 7
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 7 + 3 + 1 = 11
- Rectangles (including squares):
- 1×1 rectangles: 4
- 1×2 rectangles: 3
- 2×1 rectangles: 3
- 2×2 rectangles: 1
Total number of rectangles: 4 + 3 + 3 + 1 = 11
Total number of distinct shapes (triangles and rectangles): 11 + 11 = 22
Example 9: Counting Circles and Semi-Circles
Question: How many circles and semi-circles are there in the figure below?
_ _
_( )_
/ \
\ _ _ _ _ /
Answer:
- Full circles: 1
- Semi-circles: 2
Total number of circles and semi-circles: 1 + 2 = 3
Example 10: Counting Lines in a Star
Question: How many straight lines are there in the star figure below?
/\
/__\
\/\
/__\/
\ /
\/
Answer:
- Vertical lines: 3
- Horizontal lines: 2
- Diagonal lines: 6
Total number of straight lines: 3 + 2 + 6 = 11
Example 11: Counting Diagonals in a Polygon
Question: How many diagonals are there in a hexagon?
____
/ \
\_____/
Answer:
- Number of vertices (n): 6
- Number of diagonals (formula): n(n−3)2\frac{n(n-3)}{2}
Total number of diagonals: 6(6−3)2=6×32=9\frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9
Example 12: Counting Intersecting Lines in a Circle
Question: How many intersecting lines are there if 5 points are connected inside a circle?
O
/|\
/ | \
/ | \
/___|___\
Answer:
- Number of points (n): 5
- Number of intersecting lines (combination formula): (n2)=n(n−1)2\binom{n}{2} = \frac{n(n-1)}{2}
Total number of intersecting lines: (52)=5(5−1)2=5×42=10
Example 13: Counting Shapes in a Complex Geometric Pattern
Question: How many shapes (triangles, squares, and rectangles) are there in the figure below?
/\
/__\
/\ /\
\/__\/
Answer:
- Triangles:
- Small triangles: 5
- Medium triangles: 2
- Large triangles: 1
Total number of triangles: 5 + 2 + 1 = 8
- Rectangles (including squares):
- 1×1 rectangles: 3
- 1×2 rectangles: 2
- 2×1 rectangles: 2
- 2×2 rectangles: 1
Total number of rectangles: 3 + 2 + 2 + 1 = 8
Example 14: Counting Triangles in an Equilateral Triangle
Question: How many triangles are there in an equilateral triangle divided into smaller equilateral triangles?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 9
- Medium triangles: 4
- Large triangles: 1
Total number of triangles: 9 + 4 + 1 = 14
Example 15: Counting Overlapping Circles
Question: How many regions are there when three circles overlap each other?
O
/ \
O O
\ /
O
Answer:
- Individual circles: 3
- Overlapping regions: 4
Total number of regions: 3 + 4 = 7
Example 16: Counting Intersecting Lines in a Star
Question: How many intersecting lines are there in a five-pointed star?
/\
/__\
\/\
/__\
\ /
\/
Answer:
- Vertical lines: 2
- Horizontal lines: 2
- Diagonal lines: 5
Total number of intersecting lines: 2 + 2 + 5 = 9
Example 17: Counting Shapes in a Nested Grid
Question: How many squares and rectangles are there in a 5×5 nested grid?
_ _ _ _ _
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 25
- 2×2 squares: 16
- 3×3 squares: 9
- 4×4 squares: 4
- 5×5 square: 1
Total number of squares: 25 + 16 + 9 + 4 + 1 = 55
- Rectangles (including squares):
- 1×1 rectangles: 25
- 1×2 rectangles: 20
- 1×3 rectangles: 15
- 1×4 rectangles: 10
- 1×5 rectangles: 5
- 2×1 rectangles: 20
- 2×2 rectangles: 16
- 2×3 rectangles: 12
- 2×4 rectangles: 8
- 2×5 rectangles: 4
- 3×1 rectangles: 15
- 3×2 rectangles: 12
- 3×3 rectangles: 9
- 3×4 rectangles: 6
- 3×5 rectangles: 3
- 4×1 rectangles: 10
- 4×2 rectangles: 8
- 4×3 rectangles: 6
- 4×4 rectangles: 4
- 4×5 rectangles: 2
- 5×1 rectangles: 5
- 5×2 rectangles: 4
- 5×3 rectangles: 3
- 5×4 rectangles: 2
- 5×5 rectangle: 1
Total number of rectangles (including squares): 25 + 20 + 15 + 10 + 5 + 20 + 16 + 12 + 8 + 4 + 15 + 12 + 9 + 6 + 3 + 10 + 8 + 6 + 4 + 2 + 5 + 4 + 3 + 2 + 1 = 204
Example 18: Counting Circles in Concentric Circles
Question: How many circles are there in the figure below with concentric circles?
______
/ \
/________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 19: Counting Shapes in Overlapping Squares
Question: How many distinct regions are there in two overlapping squares?
____
/ /
/___/
\____
/
/___
Answer:
- Individual regions: 5
Total number of distinct regions: 5
Example 20: Counting Shapes in a Pentagon
Question: How many diagonals are there in a pentagon?
____
/ /
/___/
/ / \
/___/___\
Answer:
- Number of vertices (n): 5
- Number of diagonals (formula): n(n−3)2\frac{n(n-3)}{2}
Total number of diagonals: 5(5−3)2=5×22=5\frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5
Example 21: Counting Shapes in an Octagon
Question: How many diagonals are there in an octagon?
_______
/ \
/_________\
\ /
\_____/
Answer:
- Number of vertices (n): 8
- Number of diagonals (formula): n(n−3)2\frac{n(n-3)}{2}
Total number of diagonals: 8(8−3)2=8×52=20\frac{8(8-3)}{2} = \frac{8 \times 5}{2} = 20
Example 22: Counting Intersection Points of Lines
Question: How many intersection points are there when four lines intersect each other in general position?
\ / \ / \
\ / \ /
\ \ /
Answer:
- Number of lines (n): 4
- Number of intersection points (combination formula): (n2)=n(n−1)2\binom{n}{2} = \frac{n(n-1)}{2}
Total number of intersection points: (42)=4(4−1)2=4×32=6
Example 23: Counting Hexagons in a Grid
Question: How many hexagons are there in a 3×3 grid of smaller hexagons?
/ \ / \ / \
/___\/___\/___\
\ / \ / \ /
\/___\/___\/
/ \ / \ / \
/___\/___\/___\
\ / \ / \ /
\/___\/___\/
Answer:
- Small hexagons: 9
- Medium hexagons (formed by 2 small hexagons): 3
- Large hexagon (formed by 4 small hexagons): 1
Total number of hexagons: 9 + 3 + 1 = 13
Example 24: Counting Parallelograms in a Grid
Question: How many parallelograms are there in a 3×3 grid of smaller squares?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- 1×1 squares (considered as parallelograms): 9
- 1×2 rectangles (considered as parallelograms): 6
- 2×1 rectangles (considered as parallelograms): 6
- 2×2 squares (considered as parallelograms): 4
Total number of parallelograms: 9 + 6 + 6 + 4 = 25
Example 25: Counting Overlapping Circles in a Venn Diagram
Question: How many regions are there in a Venn diagram with three overlapping circles?
O
/ \
O O O
\ / \
O O
/ \ / \
O O O O
\ / \ / \ /
O O O
Answer:
- Individual regions: 7
Total number of regions: 7
Example 26: Counting Diagonals in a Decagon
Question: How many diagonals are there in a decagon?
_______
/ \
/_________\
| |
|_________|
Answer:
- Number of vertices (n): 10
- Number of diagonals (formula): n(n−3)2\frac{n(n-3)}{2}
Total number of diagonals: 10(10−3)2=10×72=35\frac{10(10-3)}{2} = \frac{10 \times 7}{2} = 35
Example 27: Counting Intersection Points of Lines in a Hexagon
Question: How many intersection points are there when six lines intersect each other in general position?
\ / \ / \
\/___\/
Answer:
- Number of lines (n): 6
- Number of intersection points (combination formula): (n2)=n(n−1)2\binom{n}{2} = \frac{n(n-1)}{2}
Total number of intersection points: (62)=6(6−1)2=6×52=15
Example 28: Counting Squares in a Diamond Shape
Question: How many squares are there in the diamond shape below?
_
/ \
/___\
\ /
\_/
Answer:
- Small squares: 1
- Medium squares (formed by 2 small squares): 0
- Large squares (formed by 4 small squares): 0
Total number of squares: 1
Example 29: Counting Shapes in a Ladder Shape
Question: How many triangles and rectangles are there in the ladder shape below?
____
/ /
|___|
| |
|___|
Answer:
- Triangles:
- Small triangles: 0
- Medium triangles: 0
- Large triangles: 0
Total number of triangles: 0
- Rectangles (including squares):
- 1×1 rectangles: 2
- 1×2 rectangles: 1
- 2×1 rectangles: 1
- 2×2 rectangles: 0
Total number of rectangles: 2 + 1 + 1 = 4
Example 30: Counting Overlapping Rectangles
Question: How many rectangles are there in the overlapping figure below?
____
| |
|____|
| |
|____|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 31: Counting Triangles in a Hexagon
Question: How many triangles are there in a hexagon divided into smaller triangles?
_______
/ \ / \ / \
\_/ \_/ \_/
Answer:
- Small triangles: 6
- Medium triangles: 2
- Large triangles: 1
Total number of triangles: 6 + 2 + 1 = 9
Example 32: Counting Shapes in a Cross Shape
Question: How many triangles and rectangles are there in the cross shape below?
_
| |
|_|
| |
|_|
Answer:
- Triangles:
- Small triangles: 0
- Medium triangles: 0
- Large triangles: 0
Total number of triangles: 0
- Rectangles (including squares):
- 1×1 rectangles: 3
- 1×2 rectangles: 2
- 2×1 rectangles: 2
- 2×2 rectangles: 1
Total number of rectangles: 3 + 2 + 2 + 1 = 8
Example 33: Counting Shapes in a Triangular Grid
Question: How many triangles and rectangles are there in a triangular grid?
/\
/__\
/\ /\
/__\/__\
Answer:
- Triangles:
- Small triangles: 4
- Medium triangles: 2
- Large triangles: 1
Total number of triangles: 4 + 2 + 1 = 7
- Rectangles:
- Small rectangles: 0
- Medium rectangles: 0
- Large rectangles: 0
Total number of rectangles: 0
Example 34: Counting Shapes in a Hexagonal Grid
Question: How many hexagons are there in a 2×2 grid of smaller hexagons?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
Answer:
- Small hexagons: 4
- Medium hexagons (formed by 2 small hexagons): 2
- Large hexagons (formed by 4 small hexagons): 1
Total number of hexagons: 4 + 2 + 1 = 7
Example 35: Counting Shapes in a Rhombus
Question: How many triangles and rhombuses are there in a figure formed by dividing a rhombus into smaller triangles?
/\
/__\
\ /
\/
Answer:
- Triangles:
- Small triangles: 4
- Medium triangles: 2
- Large triangles: 1
Total number of triangles: 4 + 2 + 1 = 7
- Rhombuses:
- Small rhombuses: 1
- Medium rhombuses: 0
- Large rhombuses: 0
Total number of rhombuses: 1
Example 36: Counting Shapes in a Circular Grid
Question: How many circles and semi-circles are there in a figure formed by dividing a circle into smaller sections?
_______
/ \
\_______/
Answer:
- Circles:
- Small circles: 2
- Medium circles: 1
- Large circles: 1
Total number of circles: 2 + 1 + 1 = 4
- Semi-circles:
- Small semi-circles: 0
- Medium semi-circles: 0
- Large semi-circles: 0
Total number of semi-circles: 0
Example 37: Counting Shapes in a Star Grid
Question: How many stars are there in a grid formed by arranging stars?
*
* *
* * *
* * * *
Answer:
- Small stars: 4
- Medium stars (formed by 2 small stars): 3
- Large stars (formed by 4 small stars): 2
Total number of stars: 4 + 3 + 2 = 9
Example 38: Counting Circles in Overlapping Rings
Question: How many circles are there in the figure below with three overlapping rings?
OOO
O O
O O
O O
OOO
Answer:
- Total rings: 3
- Overlapping regions: 4
Total number of circles: 3 + 4 = 7
Example 39: Counting Triangles in a Fractal Pattern
Question: How many triangles are there in a fractal pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
/ /\ /\ /\
/__/__\/__\/__\
Answer:
- Small triangles: 16
- Medium triangles: 8
- Large triangles: 4
- Extra-large triangles: 2
- Largest triangle: 1
Total number of triangles: 16 + 8 + 4 + 2 + 1 = 31
Example 40: Counting Rectangles in Nested Squares
Question: How many rectangles are there in a nested squares pattern?
__ __ __
|__|__|__|
|__|__|__|
|__|__|__|
Answer:
- Small rectangles: 9
- Medium rectangles: 4
- Large rectangles: 1
Total number of rectangles: 9 + 4 + 1 = 14
Example 41: Counting Intersecting Lines in a Grid
Question: How many intersecting lines are there in a 3×3 grid?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- Horizontal lines: 3
- Vertical lines: 3
- Intersecting points: 9
Total number of intersecting lines: 3 + 3 + 9 = 15
Example 42: Counting Regions in a Divided Square
Question: How many regions are there in a square divided into smaller sections?
__ __
| | |
|__|__|
| | |
|__|__|
Answer:
- Small squares: 4
- Medium squares: 2
- Large square: 1
Total number of regions: 4 + 2 + 1 = 7
Example 43: Counting Circles and Semi-Circles in Overlapping Rings
Question: How many circles and semi-circles are there in the figure below with overlapping rings?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Semi-circles: 2
Total number of circles and semi-circles: 3 + 2 = 5
Example 44: Counting Triangles in a Sierpinski Triangle
Question: How many triangles are there in a Sierpinski triangle of order 2?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
Answer:
- Small triangles: 9
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 9 + 3 + 1 = 13
Example 45: Counting Rectangles in a Grid with Overlapping Lines
Question: How many rectangles are there in a grid with overlapping lines?
__ __
|__|__|
|__|__|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 46: Counting Intersecting Lines in a Hexagonal Grid
Question: How many intersecting lines are there in a hexagonal grid?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
\ / \ /
\/___\/
Answer:
- Horizontal lines: 3
- Vertical lines: 3
- Diagonal lines: 6
Total number of intersecting lines: 3 + 3 + 6 = 12
Example 47: Counting Regions in a Square with Diagonal Lines
Question: How many regions are there in a square divided by diagonal lines?
__ __
| / /
|/ /
|\ \
|__\__\
Answer:
- Small regions: 4
- Medium regions: 2
- Large regions: 1
Total number of regions: 4 + 2 + 1 = 7
Example 48: Counting Shapes in a 3×3 Hexagonal Grid
Question: How many hexagons are there in a 3×3 hexagonal grid?
/ \ / \ / \
/___\/___\/___\
\ / \ / \ /
\/___\/___\/
/ \ / \ / \
/___\/___\/___\
\ / \ / \ /
\/___\/___\/
/ \ / \ / \
/___\/___\/___\
Answer:
- Small hexagons: 9
- Medium hexagons: 3
- Large hexagon: 1
Total number of hexagons: 9 + 3 + 1 = 13
Example 49: Counting Triangles in a Pyramid Shape
Question: How many triangles are there in a pyramid shape?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
Answer:
- Small triangles: 9
- Medium triangles: 4
- Large triangles: 1
Total number of triangles: 9 + 4 + 1 = 14
Example 50: Counting Rectangles in a 2×2 Grid
Question: How many rectangles are there in a 2×2 grid?
_ _
|_|_|
|_|_|
Answer:
- 1×1 rectangles: 4
- 1×2 rectangles: 4
- 2×1 rectangles: 4
- 2×2 rectangles: 1
Total number of rectangles: 4 + 4 + 4 + 1 = 13
Example 51: Counting Intersection Points in a Circle with Chords
Question: How many intersection points are there if 6 points are connected inside a circle?
O
/ \
/ | \
O / | \ O
\ | | | /
\ / \ /
O O
Answer:
- Number of points (n): 6
- Number of intersection points (combination formula): (n2)=n(n−1)2\binom{n}{2} = \frac{n(n-1)}{2}
Total number of intersection points: (62)=6(6−1)2=6×52=15\binom{6}{2} = \frac{6(6-1)}{2} = \frac{6 \times 5}{2} = 15
Example 52: Counting Shapes in a 4×4 Grid
Question: How many squares and rectangles are there in a 4×4 grid?
_ _ _ _
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 16
- 2×2 squares: 9
- 3×3 squares: 4
- 4×4 square: 1
Total number of squares: 16 + 9 + 4 + 1 = 30
- Rectangles (including squares):
- 1×1 rectangles: 16
- 1×2 rectangles: 12
- 1×3 rectangles: 8
- 1×4 rectangles: 4
- 2×1 rectangles: 12
- 2×2 rectangles: 9
- 2×3 rectangles: 6
- 2×4 rectangles: 3
- 3×1 rectangles: 8
- 3×2 rectangles: 6
- 3×3 rectangles: 4
- 3×4 rectangles: 2
- 4×1 rectangles: 4
- 4×2 rectangles: 3
- 4×3 rectangles: 2
- 4×4 rectangle: 1
Total number of rectangles (including squares): 16 + 12 + 8 + 4 + 12 + 9 + 6 + 3 + 8 + 6 + 4 + 2 + 4 + 3 + 2 + 1 = 100
Example 53: Counting Regions in a Figure with Overlapping Lines
Question: How many regions are there in a figure with multiple overlapping lines?
__ __
| / /
|/ /
|\ \
|__\__\
Answer:
- Small regions: 4
- Medium regions: 2
- Large region: 1
Total number of regions: 4 + 2 + 1 = 7
Example 54: Counting Triangles in a Diamond Pattern
Question: How many triangles are there in a diamond pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 8
- Medium triangles: 4
- Large triangles: 2
Total number of triangles: 8 + 4 + 2 = 14
Example 55: Counting Rectangles in an Overlapping Grid
Question: How many rectangles are there in an overlapping grid?
__ __ __
|__|__|__|
|__|__|__|
|__|__|__|
Answer:
- Small rectangles: 9
- Medium rectangles: 4
- Large rectangles: 1
Total number of rectangles: 9 + 4 + 1 = 14
Example 56: Counting Circles in a Pattern
Question: How many circles are there in a pattern of overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Overlapping circles: 2
Total number of circles: 3 + 2 = 5
Example 57: Counting Intersecting Lines in a Pattern
Question: How many intersecting lines are there in the pattern below?
\ | /
\ | /
\|/
--- + ---
/|\
/ | \
/ | \
Answer:
- Horizontal lines: 1
- Vertical lines: 1
- Diagonal lines: 4
Total number of intersecting lines: 1 + 1 + 4 = 6
Example 58: Counting Regions in a Figure with Overlapping Circles
Question: How many regions are there in a figure with overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Regions formed by overlapping circles: 7
Total number of regions: 7
Example 59: Counting Shapes in a Star Pattern
Question: How many triangles and rectangles are there in a star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Also read : Reasoning question for grouping of identical figures
Example 60: Counting Shapes in a 5×5 Grid
Question: How many squares and rectangles are there in a 5×5 grid?
_ _ _ _ _
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 25
- 2×2 squares: 16
- 3×3 squares: 9
- 4×4 squares: 4
- 5×5 square: 1
Total number of squares: 25 + 16 + 9 + 4 + 1 = 55
- Rectangles (including squares):
- 1×1 rectangles: 25
- 1×2 rectangles: 20
- 1×3 rectangles: 15
- 1×4 rectangles: 10
- 1×5 rectangles: 5
- 2×1 rectangles: 20
- 2×2 rectangles: 16
- 2×3 rectangles: 12
- 2×4 rectangles: 8
- 2×5 rectangles: 4
- 3×1 rectangles: 15
- 3×2 rectangles: 12
- 3×3 rectangles: 9
- 3×4 rectangles: 6
- 3×5 rectangles: 3
- 4×1 rectangles: 10
- 4×2 rectangles: 8
- 4×3 rectangles: 6
- 4×4 rectangles: 4
- 4×5 rectangles: 2
- 5×1 rectangles: 5
- 5×2 rectangles: 4
- 5×3 rectangles: 3
- 5×4 rectangles: 2
- 5×5 rectangle: 1
Total number of rectangles (including squares): 25 + 20 + 15 + 10 + 5 + 20 + 16 + 12 + 8 + 4 + 15 + 12 + 9 + 6 + 3 + 10 + 8 + 6 + 4 + 2 + 5 + 4 + 3 + 2 + 1 = 204
Example 61: Counting Shapes in a 4×4 Hexagonal Grid
Question: How many hexagons are there in a 4×4 hexagonal grid?
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
Answer:
- Small hexagons: 16
- Medium hexagons: 6
- Large hexagons: 4
- Extra-large hexagon: 1
Total number of hexagons: 16 + 6 + 4 + 1 = 27
Example 62: Counting Triangles in a Pentagon
Question: How many triangles are there in a pentagon divided into smaller triangles?
_____
/ \ / \
/ \/ \
\ /\ /
\ / \ /
Answer:
- Small triangles: 10
- Medium triangles: 5
- Large triangles: 1
Total number of triangles: 10 + 5 + 1 = 16
Example 63: Counting Rectangles in a Lattice Pattern
Question: How many rectangles are there in a lattice pattern?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- Small rectangles: 9
- Medium rectangles: 4
- Large rectangles: 1
Total number of rectangles: 9 + 4 + 1 = 14
Example 64: Counting Regions in an Overlapping Square Pattern
Question: How many regions are there in an overlapping square pattern?
_ _ _
| | | |
|_|_|_|
| | | |
|_|_|_|
Answer:
- Small regions: 4
- Medium regions: 2
- Large regions: 1
Total number of regions: 4 + 2 + 1 = 7
Example 65: Counting Shapes in a Circular Pattern
Question: How many circles and semi-circles are there in a circular pattern?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Semi-circles: 2
Total number of circles and semi-circles: 3 + 2 = 5
Example 66: Counting Triangles in a Hexagonal Grid
Question: How many triangles are there in a hexagonal grid?
/ \
/___\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
Example 67: Counting Rectangles in a Circular Pattern
Question: How many rectangles are there in a circular pattern?
____
| |
__|____|__
| |
| |
|__ ____|
|__|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 68: Counting Circles in a Pattern of Overlapping Circles
Question: How many circles are there in a pattern of overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Overlapping regions: 4
Total number of circles: 3 + 4 = 7
Example 69: Counting Intersection Points in a Complex Grid
Question: How many intersection points are there in a complex grid?
\ | /
\ | /
\|/
---- + ----
/|\
/ | \
/ | \
Answer:
- Horizontal lines: 1
- Vertical lines: 1
- Diagonal lines: 4
Total number of intersection points: 1 + 1 + 4 = 6
Example 70: Counting Regions in a Figure with Multiple Circles
Question: How many regions are there in a figure with multiple overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Regions formed by overlapping circles: 7
Total number of regions: 7
Example 71: Counting Shapes in a Star Pattern
Question: How many triangles and rectangles are there in a star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Example 72: Counting Triangles in a Concentric Triangle Pattern
Question: How many triangles are there in a concentric triangle pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
\__\/__\/
Answer:
- Small triangles: 12
- Medium triangles: 6
- Large triangles: 3
Total number of triangles: 12 + 6 + 3 = 21
Example 73: Counting Rectangles in a Nested Grid
Question: How many rectangles are there in a nested grid pattern?
_ _ _ _
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- Small rectangles: 16
- Medium rectangles: 6
- Large rectangles: 4
- Extra-large rectangles: 1
Total number of rectangles: 16 + 6 + 4 + 1 = 27
Example 74: Counting Circles in an Overlapping Circular Pattern
Question: How many circles are there in a pattern of overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Overlapping regions: 4
Total number of circles: 3 + 4 = 7
Example 75: Counting Intersection Points in a Complex Grid
Question: How many intersection points are there in a complex grid pattern?
\ | /
\ | /
\|/
---- + ----
/|\
/ | \
/ | \
Answer:
- Horizontal lines: 1
- Vertical lines: 1
- Diagonal lines: 4
Total number of intersection points: 1 + 1 + 4 = 6
Example 76: Counting Regions in a Figure with Multiple Overlapping Squares
Question: How many regions are there in a figure with multiple overlapping squares?
_ _ _ _
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- Small regions: 16
- Medium regions: 6
- Large regions: 4
- Extra-large regions: 1
Total number of regions: 16 + 6 + 4 + 1 = 27
Example 77: Counting Shapes in a Complex Star Pattern
Question: How many triangles and rectangles are there in a complex star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Example 78: Counting Shapes in a Fibonacci Pattern
Question: How many rectangles and squares are there in a Fibonacci pattern?
_ _
|_|_|
|_|_|
|_|_|
Answer:
- Squares:
- 1×1 squares: 4
- 2×2 squares: 1
Total number of squares: 4 + 1 = 5
- Rectangles (including squares):
- 1×1 rectangles: 4
- 1×2 rectangles: 3
- 2×1 rectangles: 3
- 2×2 rectangles: 1
Total number of rectangles (including squares): 4 + 3 + 3 + 1 = 11
Example 79: Counting Shapes in a Concentric Square Pattern
Question: How many squares are there in a concentric square pattern?
__ __
|__|__|
|__|__|
Answer:
- Small squares: 4
- Medium squares: 1
Total number of squares: 4 + 1 = 5
Example 80: Counting Rectangles in a Nested Rectangle Pattern
Question: How many rectangles are there in a nested rectangle pattern?
__ __
|__|__|
|__|__|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 81: Counting Circles in an Overlapping Circular Grid
Question: How many circles are there in an overlapping circular grid?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Overlapping regions: 4
Total number of circles: 3 + 4 = 7
Example 82: Counting Intersection Points in a Hexagonal Grid
Question: How many intersection points are there in a hexagonal grid?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
\ / \ /
\/___\/
Answer:
- Horizontal lines: 3
- Vertical lines: 3
- Diagonal lines: 6
Total number of intersection points: 3 + 3 + 6 = 12
Example 83: Counting Regions in a Figure with Multiple Overlapping Hexagons
Question: How many regions are there in a figure with multiple overlapping hexagons?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
\ / \ /
\/___\/
Answer:
- Regions formed by overlapping hexagons: 7
Total number of regions: 7
Example 84: Counting Shapes in a Complex Circular Pattern
Question: How many circles and semi-circles are there in a complex circular pattern?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Semi-circles: 2
Total number of circles and semi-circles: 3 + 2 = 5
Example 85: Counting Shapes in a Nested Star Pattern
Question: How many triangles and rectangles are there in a nested star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Example 86: Counting Triangles in a Fractal Pattern
Question: How many triangles are there in a fractal pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
/\ /\ /\ /\
/__\/__\/__\/__\
Answer:
- Small triangles: 16
- Medium triangles: 8
- Large triangles: 4
- Extra-large triangles: 2
- Largest triangle: 1
Total number of triangles: 16 + 8 + 4 + 2 + 1 = 31
Example 87: Counting Rectangles in a 3×3 Grid
Question: How many rectangles are there in a 3×3 grid?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- 1×1 rectangles: 9
- 1×2 rectangles: 6
- 1×3 rectangles: 3
- 2×1 rectangles: 6
- 2×2 rectangles: 4
- 2×3 rectangles: 2
- 3×1 rectangles: 3
- 3×2 rectangles: 2
- 3×3 rectangles: 1
Total number of rectangles: 9 + 6 + 3 + 6 + 4 + 2 + 3 + 2 + 1 = 36
Example 88: Counting Circles in Concentric Circles
Question: How many circles are there in the figure below with concentric circles?
_______
/ \
/_________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 89: Counting Shapes in Overlapping Squares
Question: How many distinct regions are there in two overlapping squares?
____
/ /
/___/
\____
/
/___
Answer:
- Individual regions: 5
Total number of distinct regions: 5
Example 90: Counting Triangles in a Pentagonal Pattern
Question: How many triangles are there in a pentagonal pattern?
_____
/ \ / \
/ \/ \
\ /\ /
\ / \ /
Answer:
- Small triangles: 10
- Medium triangles: 5
- Large triangles: 1
Total number of triangles: 10 + 5 + 1 = 16
Example 91: Counting Regions in a Figure with Multiple Overlapping Lines
Question: How many regions are there in a figure with multiple overlapping lines?
\ | /
\ | /
\|/
---- + ----
/|\
/ | \
/ | \
Answer:
- Regions formed by overlapping lines: 8
Total number of regions: 8
Example 92: Counting Shapes in a Star Pattern
Question: How many triangles and rectangles are there in a star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Example 93: Counting Triangles in a Sierpinski Triangle
Question: How many triangles are there in a Sierpinski triangle of order 3?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
/\ /\ /\ /\
/__\/__\/__\/__\
/\ /\ /\ /\ /\
\__\/__\/__\/__\/
Answer:
- Small triangles: 27
- Medium triangles: 9
- Large triangles: 3
- Extra-large triangle: 1
Total number of triangles: 27 + 9 + 3 + 1 = 40
Example 94: Counting Rectangles in a 4×4 Grid
Question: How many rectangles are there in a 4×4 grid?
_ _ _ _
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
|_|_|_|_|
Answer:
- 1×1 rectangles: 16
- 1×2 rectangles: 12
- 1×3 rectangles: 8
- 1×4 rectangles: 4
- 2×1 rectangles: 12
- 2×2 rectangles: 9
- 2×3 rectangles: 6
- 2×4 rectangles: 3
- 3×1 rectangles: 8
- 3×2 rectangles: 6
- 3×3 rectangles: 4
- 3×4 rectangles: 2
- 4×1 rectangles: 4
- 4×2 rectangles: 3
- 4×3 rectangles: 2
- 4×4 rectangle: 1
Total number of rectangles: 16 + 12 + 8 + 4 + 12 + 9 + 6 + 3 + 8 + 6 + 4 + 2 + 4 + 3 + 2 + 1 = 100
Example 95: Counting Circles in a Pattern of Concentric Circles
Question: How many circles are there in a pattern of concentric circles?
_______
/ \
/_________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 96: Counting Shapes in Overlapping Triangles
Question: How many distinct regions are there in two overlapping triangles?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Individual regions: 6
Total number of distinct regions: 6
Example 97: Counting Triangles in an Octagonal Pattern
Question: How many triangles are there in an octagonal pattern?
_______
/ /\ /
/__/ \/
/ /\ /\
/__/ \/__\
\ /\ / /
\/__\/__/
Answer:
- Small triangles: 14
- Medium triangles: 7
- Large triangles: 2
Total number of triangles: 14 + 7 + 2 = 23
Example 98: Counting Regions in a Figure with Multiple Overlapping Circles
Question: How many regions are there in a figure with multiple overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Regions formed by overlapping circles: 11
Total number of regions: 11
Example 99: Counting Shapes in a Nested Hexagon Pattern
Question: How many hexagons are there in a nested hexagon pattern?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
\ / \ /
\/___\/
Answer:
- Small hexagons: 4
- Medium hexagons: 2
- Large hexagon: 1
Total number of hexagons: 4 + 2 + 1 = 7
Example 100: Counting Shapes in an Overlapping Circular Pattern
Question: How many circles and semi-circles are there in an overlapping circular pattern?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Semi-circles: 2
Total number of circles and semi-circles: 3 + 2 = 5
I hope these additional examples are helpful! If you need more examples or have any specific requests, feel free to let me know. I’m here to assist!
more
Of course! Here are even more examples of counting figures:
Example 101: Counting Triangles in a Pyramid Pattern
Question: How many triangles are there in a pyramid pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
Answer:
- Small triangles: 9
- Medium triangles: 6
- Large triangles: 3
Total number of triangles: 9 + 6 + 3 = 18
Example 102: Counting Squares in a 5×5 Grid
Question: How many squares are there in a 5×5 grid?
_ _ _ _ _
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
Answer:
- 1×1 squares: 25
- 2×2 squares: 16
- 3×3 squares: 9
- 4×4 squares: 4
- 5×5 square: 1
Total number of squares: 25 + 16 + 9 + 4 + 1 = 55
Example 103: Counting Circles in a Figure
Question: How many circles are there in the figure below with three concentric circles?
_______
/ \
/_________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 104: Counting Rectangles in a 3×2 Grid
Question: How many rectangles are there in a 3×2 grid?
_ _
|_|_|
|_|_|
|_|_|
Answer:
- 1×1 rectangles: 6
- 1×2 rectangles: 3
- 2×1 rectangles: 4
- 2×2 rectangles: 2
- 3×1 rectangles: 2
- 3×2 rectangles: 1
Total number of rectangles: 6 + 3 + 4 + 2 + 2 + 1 = 18
Example 105: Counting Triangles in an Octagon
Question: How many triangles are there in an octagon divided into smaller triangles?
_______
/ /\ /
/__/ \/
\ /\ /
\/__\/
Answer:
- Small triangles: 8
- Medium triangles: 4
- Large triangles: 2
Total number of triangles: 8 + 4 + 2 = 14
Example 106: Counting Regions in a Figure with Overlapping Squares
Question: How many regions are there in a figure with overlapping squares?
__ __
/ /
|___/
\ __
/ \
/______\
Answer:
- Small regions: 5
- Medium regions: 2
- Large regions: 1
Total number of regions: 5 + 2 + 1 = 8
Example 107: Counting Shapes in a Nested Triangle Pattern
Question: How many triangles are there in a nested triangle pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
\__\/__\/
Answer:
- Small triangles: 12
- Medium triangles: 6
- Large triangles: 3
Total number of triangles: 12 + 6 + 3 = 21
Example 108: Counting Triangles in a Diamond Pattern
Question: How many triangles are there in a diamond pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 8
- Medium triangles: 4
- Large triangles: 2
Total number of triangles: 8 + 4 + 2 = 14
Example 109: Counting Rectangles in a Nested Rectangle Pattern
Question: How many rectangles are there in a nested rectangle pattern?
__ __
|__|__|
|__|__|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 110: Counting Circles in a Pattern of Concentric Circles
Question: How many circles are there in a pattern of concentric circles?
_______
/ \
/_________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 111: Counting Shapes in Overlapping Squares
Question: How many distinct regions are there in two overlapping squares?
____
/ /
/___/
\____
/
/___
Answer:
- Individual regions: 5
Total number of distinct regions: 5
Example 112: Counting Triangles in a Pentagonal Pattern
Question: How many triangles are there in a pentagonal pattern?
_____
/ \ / \
/ \/ \
\ /\ /
\ / \ /
Answer:
- Small triangles: 10
- Medium triangles: 5
- Large triangles: 1
Total number of triangles: 10 + 5 + 1 = 16
Example 113: Counting Regions in a Figure with Multiple Overlapping Lines
Question: How many regions are there in a figure with multiple overlapping lines?
\ | /
\ | /
\|/
---- + ----
/|\
/ | \
/ | \
Answer:
- Regions formed by overlapping lines: 8
Total number of regions: 8
Example 114: Counting Shapes in a Star Pattern
Question: How many triangles and rectangles are there in a star pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Triangles:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
- Rectangles:
- Small rectangles: 3
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 3 + 2 + 1 = 6
Example 115: Counting Shapes in a Hexagonal Grid
Question: How many hexagons are there in a 2×2 grid of smaller hexagons?
/ \ / \
/___\/___\
\ / \ /
\/___\/
/ \ / \
/___\/___\
Answer:
- Small hexagons: 4
- Medium hexagons: 2
- Large hexagons: 1
Total number of hexagons: 4 + 2 + 1 = 7
Example 116: Counting Shapes in an Overlapping Rectangle Pattern
Question: How many rectangles are there in an overlapping rectangle pattern?
____
| |
|____|
| |
|____|
Answer:
- Small rectangles: 4
- Medium rectangles: 2
- Large rectangles: 1
Total number of rectangles: 4 + 2 + 1 = 7
Example 117: Counting Triangles in a Complex Grid
Question: How many triangles are there in a complex grid?
/\
/__\
/\ /\
/__\/__\
/\ /\
/__\/__\
Answer:
- Small triangles: 9
- Medium triangles: 4
- Large triangles: 2
Total number of triangles: 9 + 4 + 2 = 15
Example 118: Counting Circles in an Overlapping Circle Pattern
Question: How many circles are there in a pattern of overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Full circles: 3
- Overlapping regions: 4
Total number of circles: 3 + 4 = 7
Example 119: Counting Shapes in a Star Grid
Question: How many triangles and squares are there in a star grid?
* *
* * * *
* * *
* * * *
* *
Answer:
- Triangles:
- Small triangles: 8
- Medium triangles: 4
- Large triangles: 2
Total number of triangles: 8 + 4 + 2 = 14
- Squares:
- Small squares: 4
- Medium squares: 2
- Large squares: 1
Total number of squares: 4 + 2 + 1 = 7
Example 120: Counting Regions in a Figure with Overlapping Circles
Question: How many regions are there in a figure with overlapping circles?
OOO
O O
O O
O O
OOO
Answer:
- Regions formed by overlapping circles: 7
Total number of regions: 7
Example 121: Counting Triangles in a Nested Triangle Pattern
Question: How many triangles are there in a nested triangle pattern?
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
\__\/__\/
Answer:
- Small triangles: 12
- Medium triangles: 6
- Large triangles: 3
Total number of triangles: 12 + 6 + 3 = 21
Example 122: Counting Rectangles in a Complex Grid
Question: How many rectangles are there in a complex grid?
_ _ _
|_|_|_|
|_|_|_|
|_|_|_|
|_|_|_|
Answer:
- Small rectangles: 16
- Medium rectangles: 6
- Large rectangles: 4
- Extra-large rectangles: 1
Total number of rectangles: 16 + 6 + 4 + 1 = 27
Example 123: Counting Circles in a Pattern of Concentric Circles
Question: How many circles are there in a pattern of concentric circles?
_______
/ \
/_________\
Answer:
- Concentric circles: 3
Total number of circles: 3
Example 124: Counting Triangles in an Overlapping Triangle Pattern
Question: How many triangles are there in an overlapping triangle pattern?
/\
/__\
/\ /\
/__\/__\
\ /\ /
\/__\/
Answer:
- Small triangles: 6
- Medium triangles: 3
- Large triangles: 1
Total number of triangles: 6 + 3 + 1 = 10
Example 125: Counting Shapes in a 4×4 Hexagonal Grid
Question: How many hexagons are there in a 4×4 hexagonal grid?
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
Answer:
- Small hexagons: 16
- Medium hexagons: 6
- Large hexagons: 4
- Extra-large hexagon: 1
Total number of hexagons: 16 + 6 + 4 + 1 = 27
Example 126: Counting Shapes in a 5×5 Grid
Question: How many squares and rectangles are there in a 5×5 grid?
_ _ _ _ _
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
|_|_|_|_|_|
Answer:
- Squares:
- 1×1 squares: 25
- 2×2 squares: 16
- 3×3 squares: 9
- 4×4 squares: 4
- 5×5 square: 1
Total number of squares: 25 + 16 + 9 + 4 + 1 = 55
- Rectangles (including squares):
- 1×1 rectangles: 25
- 1×2 rectangles: 20
- 1×3 rectangles: 15
- 1×4 rectangles: 10
- 1×5 rectangles: 5
- 2×1 rectangles: 20
- 2×2 rectangles: 16
- 2×3 rectangles: 12
- 2×4 rectangles: 8
- 2×5 rectangles: 4
- 3×1 rectangles: 15
- 3×2 rectangles: 12
- 3×3 rectangles: 9
- 3×4 rectangles: 6
- 3×5 rectangles: 3
- 4×1 rectangles: 10
- 4×2 rectangles: 8
- 4×3 rectangles: 6
- 4×4 rectangles: 4
- 4×5 rectangles: 2
- 5×1 rectangles: 5
- 5×2 rectangles: 4
- 5×3 rectangles: 3
- 5×4 rectangles: 2
- 5×5 rectangle: 1
Total number of rectangles (including squares): 25 + 20 + 15 + 10 + 5 + 20 + 16 + 12 + 8 + 4 + 15 + 12 + 9 + 6 + 3 + 10 + 8 + 6 + 4 + 2 + 5 + 4 + 3 + 2 + 1 = 204
Example 127: Counting Shapes in a 4×4 Hexagonal Grid
Question: How many hexagons are there in a 4×4 hexagonal grid?
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
/ \ / \ / \ / \
/___\/___\/___\/___\
\ / \ / \ / \ /
\/___\/___\/___\/
Answer:
- Small hexagons: 16
- Medium hexagons: 6
- Large hexagons: 4
- Extra-large hexagon: 1
Total number of hexagons: 16 + 6 + 4 + 1 = 27
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