Question 1
Count the total number of squares in the 2x2 grid below:
+---+---+
| | |
+---+---+
| | |
+---+---+
In a 2x2 grid: 4 + 1 = 5
Counting Figures - Advanced Level: triangle counting ADVANCED
Master counting figures concepts through this hard problem set practice set. Worksheet 26 of 30 contains 20 advanced-level problems. Deep dive into triangle counting while learning figure identification, shape counting, image enumeration. Recommended for advanced learners aiming for complex scenarios and multi-step problems.
What you'll learn in this worksheet:
- Quickly grasp figure identification essential concepts
- Learn proven shortcuts for shape counting
- Boost your hard problem set solving speed
- Spot image enumeration patterns instantly
- Achieve triangle counting mastery in minutes
Your progress through Counting Figures
Worksheet 26 of 30 (86% complete)
Question 2
Count the total number of parallelograms in the grid below (4 horizontal lines × 4 vertical lines):
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
Choose 2 horizontal lines from 4: C(4,2) = 6, Choose 2 vertical lines from 4: C(4,2) = 6. Total = 6 × 6 = 36
Question 3
Count the number of triangles in the following figure:
/\
/ \
/ \
/ \
/ \
/ \
\ /
\ /
\ /
\ /
\ /
\/
(A 6-pointed star formed by two overlapping triangles)
Tip: Count all triangles - 6 small at the points + 2 large overlapping = 8 triangles.
A hexagram contains 8 triangles: 6 small triangles at the points and 2 large overlapping triangles.
Question 4
Count the total number of triangles in the triangular grid below (1 row):
/\
/ \
/ \
/______\
Upward pointing triangles: 1, Downward pointing triangles: 0. Total = 1 + 0 = 1
Question 5
Count the total number of rectangles (including squares) in the 2x3 grid below:
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
Number of ways to choose 2 horizontal lines = C(3,2) = 3, Number of ways to choose 2 vertical lines = C(4,2) = 6. Total = 3 × 6 = 18
Question 6
Count the number of triangles in the following figure:
* *
\ /
*
/ \
* *
(A 5-pointed star with intersecting lines)
Tip: Count all triangles formed by the intersecting lines.
There are 5 small triangles at the points and 5 larger triangles inside.
A pentagram contains 10 triangles: 5 small triangles at the points and 5 larger triangles formed by intersecting lines.
Question 7
Count the number of triangles in the following figure:
/\
/|\
/ | \
/ | \
/ | \
/____|____\
\ | /
\ | /
\ | /
\ | /
\|/
The triangle has 3 cevians (lines from vertices to opposite sides). These create 6 smaller triangles within the large triangle. Total = 6 triangles.
Question 8
Count the number of line segments in the following figure:
o---o---o---o
With 4 points on a line, number of segments = C(4,2) = 6 segments.
Question 9
Count the number of triangles in the following figure:
* *
\ /
*
/ \
* *
(A 5-pointed star with intersecting lines)
Tip: Count all triangles formed by the intersecting lines.
There are 5 small triangles at the points and 5 larger triangles inside.
A pentagram contains 10 triangles: 5 small triangles at the points and 5 larger triangles formed by intersecting lines.
Question 10
Count the total number of rectangles (including squares) in the 3x4 grid below:
+---+---+---+---+
| | | | |
+---+---+---+---+
| | | | |
+---+---+---+---+
| | | | |
+---+---+---+---+
Number of ways to choose 2 horizontal lines = C(4,2) = 6, Number of ways to choose 2 vertical lines = C(5,2) = 10. Total = 6 × 10 = 60
Question 11
Count the number of triangles in the following figure:
* *
\ /
*
/ \
* *
(A 5-pointed star with intersecting lines)
Tip: Count all triangles formed by the intersecting lines.
There are 5 small triangles at the points and 5 larger triangles inside.
A pentagram contains 10 triangles: 5 small triangles at the points and 5 larger triangles formed by intersecting lines.
Question 12
Count the number of triangles in the following figure:
/\
/ \
/ \
/ \
/ \
/ \
\ /
\ /
\ /
\ /
\ /
\/
(A 6-pointed star formed by two overlapping triangles)
Tip: Count all triangles - 6 small at the points + 2 large overlapping = 8 triangles.
A hexagram contains 8 triangles: 6 small triangles at the points and 2 large overlapping triangles.
Question 13
Count the number of triangles in the following figure:
* *
\ /
*
/ \
* *
(A 5-pointed star with intersecting lines)
Tip: Count all triangles formed by the intersecting lines.
There are 5 small triangles at the points and 5 larger triangles inside.
A pentagram contains 10 triangles: 5 small triangles at the points and 5 larger triangles formed by intersecting lines.
Question 14
Count the total number of rectangles (including squares) in the 3x3 grid below:
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
Number of ways to choose 2 horizontal lines = C(4,2) = 6, Number of ways to choose 2 vertical lines = C(4,2) = 6. Total = 6 × 6 = 36
Question 15
Count the total number of triangles in the triangular grid below (2 rows):
/\
/__\
/\ /\
/__\/__\
Upward pointing triangles: 4, Downward pointing triangles: 1. Total = 4 + 1 = 5
Question 16
Count the total number of squares in the 1x1 grid below:
+---+
| |
+---+
In a 1x1 grid: 1 = 1
Question 17
Count the number of triangles in the following figure:
* *
\ /
*
/ \
* *
(A 5-pointed star with intersecting lines)
Tip: Count all triangles formed by the intersecting lines.
There are 5 small triangles at the points and 5 larger triangles inside.
A pentagram contains 10 triangles: 5 small triangles at the points and 5 larger triangles formed by intersecting lines.
Question 18
Count the number of triangles in the following figure:
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
In a 3-row triangular grid: Small triangles (9) + medium triangles formed by 4 small triangles (3) + large triangle (1) = 13 total.
Question 19
Count the total number of squares in the 2x3 grid below:
+---+---+---+
| | | |
+---+---+---+
| | | |
+---+---+---+
For a 2x3 grid: (2×3) + (1×2) = 8
Question 20
Count the total number of rectangles (including squares) in the 2x2 grid below:
+---+---+
| | |
+---+---+
| | |
+---+---+
Number of ways to choose 2 horizontal lines = C(3,2) = 3, Number of ways to choose 2 vertical lines = C(3,2) = 3. Total = 3 × 3 = 9
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