Counting Figures Exit
Logical reasoning

Counting Figures

Count systematically by size: all 1-piece shapes, then all 2-piece shapes, and so on. Or use the formulas below for standard grids.

Start with this

How many triangles are in a triangle with three lines drawn from the top? Not 4. Count by size, and it is 10.

Counting figures looks visual but it is really counting in a system. Label every point and count shapes by how many small pieces they use.

Label, then count by size

Label every corner and meeting point with a letter. Count the smallest shapes first, then shapes made of 2 pieces, 3 pieces and so on.

Triangle split by n lines from the top corner: the base has n + 1 pieces, and the number of triangles = (n + 1)(n + 2)/2.

PQRSTA
Count by size: 4 single pieces, 3 made of two, 2 of three, 1 of four = 10
A 3 × 3 grid has 9 + 4 + 1 = 14 squares

Label, then count by size: the rules

Lines from the apex splitting the base into k parts
Triangles = k(k + 1)/2
Squares in an n × n grid
12 + 22 + ... + n2
Rectangles in an m × n grid
(m+1)C2 × (n+1)C2

Worked example

A triangle has 3 lines drawn from the top corner to the base (4 small triangles). How many triangles in total?

  1. Base split into 4 parts.
    Why: Each triangle is picked by choosing two points on the base.
  2. Choose 2 of the 5 base points: 5C2 = 10.
    Why: Every pair of base points with the apex makes a triangle.
  3. 10.

Answer: 10

Worked example

How many squares are there on a chessboard (8 × 8)?

  1. 1 × 1: 64, 2 × 2: 49, 3 × 3: 36 ... 8 × 8: 1.
    Why: An n × n grid has (n - k + 1)2 squares of size k.
  2. 1 + 4 + 9 + ... + 64 = 204.

Answer: 204

Shortcut: Squares in a chessboard

Use it when: 8 × 8.

  1. 1 + 4 + 9 + ... + 64 = 204.

Squares in a chessboard: try it

Squares on a chessboard?

  1. 8 × 9 × 17 / 6.

Answer: 204

Your turn medium

Rectangles (including squares) in a 3 × 3 grid?

  1. C(4,2) × C(4,2) = 6 × 6.

Answer: 36

Your turn medium

Rectangles in a 2 × 4 grid?

  1. C(3,2) × C(5,2) = 3 × 10.

Answer: 30

Your turn medium

A triangle has 4 lines drawn from the top vertex to the base (including the two sides, so 4 lines total). Triangles?

  1. Choose 2 of the 4 lines: C(4,2).

Answer: 6

Your turn hard

Squares in a 5 × 5 grid?

  1. 1 + 4 + 9 + 16 + 25.

Answer: 55

Recap

Now practise

7 questions with full solutions.

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