ReasoningAbility.com

Counting Figures - Expert Level: circle counting EXPERT

Strategic basic drills ★ for counting figures: 20 expert-level problems. Worksheet 29 of 30 - Focus: circle counting. Develop expertise in figure counting, shape enumeration, geometric count with step-by-step solutions. Ideal for expert-level learners targeting challenging problems and time-bound practice.

📝 Worksheet 29 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Expert level

What you'll learn in this worksheet:
Your progress through Counting Figures
Worksheet 29 of 30 (96% complete)

Question 1

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\ \ | / \ | / \|/
Two medians create 4 smaller triangles. Total = 4 triangles.
triangle, basic

Question 2

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
rectangle, grid

Question 3

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.
triangle, star

Question 4

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
triangle, grid

Question 5

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\
The median divides the triangle into 2 smaller triangles. Total = 2 triangles.
triangle, basic

Question 6

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.
triangle, star

Question 7

Count the total number of parallelograms in the grid below (3 horizontal lines × 3 vertical lines): +---+---+ | | | +---+---+ | | | +---+---+
Choose 2 horizontal lines from 3: C(3,2) = 3, Choose 2 vertical lines from 3: C(3,2) = 3. Total = 3 × 3 = 9
parallelogram, grid

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.
line, segment

Question 9

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.
line, segment

Question 10

Count the total number of squares in the 2x3 grid below: +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
For a 2x3 grid: (2×3) + (1×2) = 8
square, rectangle

Question 11

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\ \ | / \ | / \|/
Two medians create 4 smaller triangles. Total = 4 triangles.
triangle, basic

Question 12

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.
triangle, composite

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.
triangle, star

Question 14

Count the number of line segments in the following figure: o-------o-------o
With 3 points on a line, number of segments = C(3,2) = 3 segments.
line, segment

Question 15

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.
triangle, star

Question 16

Count the number of triangles in the following figure: Triangular grid with 4 rows
In a 4-row triangular grid: Small triangles (16) + medium triangles of size 2 (7) + medium triangles of size 3 (3) + large triangle (1) = 27 total.
triangle, composite

Question 17

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\ \ | / \ | / \|/
Two medians create 4 smaller triangles. Total = 4 triangles.
triangle, basic

Question 18

Count the total number of squares in the 2x3 grid below: +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
For a 2x3 grid: (2×3) + (1×2) = 8
square, rectangle

Question 19

Count the total number of triangles in the triangular grid below (3 rows): /\ /__\ /\ /\ /__\/__\ /\ /\ /\ /__\/__\/__\
Upward pointing triangles: 9, Downward pointing triangles: 3. Total = 9 + 3 = 12
triangle, grid

Question 20

Count the number of line segments in the following figure: +---+ |\ /| | X | |/ \| +---+
Square has 4 sides + 2 diagonals = 6. Plus the segments created by diagonal intersection? Actually, each diagonal is split into 2 segments, giving 4 more. Total = 8 line segments.
line, segment
Previous Worksheet Next Worksheet