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Counting Figures - Intermediate Level: rectangle counting INTERMEDIATE

Exam-focused quick response training ★ worksheet: 20 intermediate-level counting figures questions. Worksheet 13 of 30 targets rectangle counting. Build proficiency in shape counting, image enumeration, design counting with detailed solutions. Ideal for mid-level competitive exam preparation.

📝 Worksheet 13 of 30 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Counting Figures
Worksheet 13 of 30 (43% complete)

Question 1

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
rectangle, grid

Question 2

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 3

Count the total number of squares in the 3x5 grid below: +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+
For a 3x5 grid: (3×5) + (2×4) + (1×3) = 26
square, rectangle

Question 4

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

Question 5

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 6

Count the number of triangles in the following figure: /\ / \ / \ /______\
There is exactly 1 triangle in the figure.
triangle, basic

Question 7

Count the number of line segments in the following figure: o-o-o-o-o
With 5 points on a line, number of segments = C(5,2) = 10 segments.
line, segment

Question 8

Count the total number of rectangles (including squares) in the 2x4 grid below: +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+
Number of ways to choose 2 horizontal lines = C(3,2) = 3, Number of ways to choose 2 vertical lines = C(5,2) = 10. Total = 3 × 10 = 30
rectangle, grid

Question 9

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
parallelogram, grid

Question 10

Count the number of squares in the following figure: A 3x3 square grid with both diagonals in the center square only: +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+ | | | | +---+---+---+
Total squares in a 3x3 grid = 14 (9 small + 4 medium + 1 large). The diagonals in the center square do not create new squares, only triangles. Total remains 14.
square, diagonals

Question 11

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 12

Count the number of triangles in the following figure: /\ /|\ / | \ /__|__\
Altitude creates 2 right triangles. Total = 2 triangles.
triangle, basic

Question 13

Count the total number of squares in the 3x5 grid below: +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+ | | | | | | +---+---+---+---+---+
For a 3x5 grid: (3×5) + (2×4) + (1×3) = 26
square, rectangle

Question 14

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 15

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 16

Count the number of triangles in the following figure: /\ /__\ /\ /\ /__\/__\
This creates 4 small congruent triangles. Total = 4 triangles.
triangle, basic

Question 17

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 18

Count the number of circles in the following figure: ___ ___ / \ / \ | \ / | | X | | / \ | \___/ \___/
Left circle (1), Right circle (1), Overlapping region (1) → Total 3 distinct circular regions.
circle, overlap

Question 19

Count the total number of squares in the 4x4 grid below: +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+ | | | | | +---+---+---+---+
In a 4x4 grid: 16 + 9 + 4 + 1 = 30
square, grid

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
rectangle, grid
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