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Concept 6 of 9

Figure Counting: Triangles & Squares

1

Today's big idea

Count the smallest shapes, then the ones built from several, then the whole outer shape.

Overlapping lines hide bigger shapes, so keep looking after the smallest ones.

2

How it works

👀 What's going on?

A drawing looks as though it holds four triangles. Count again, this time including the ones made of several pieces joined together, and the number climbs.

💡 Here's the trick

Count BIG and small shapes1234→4 small+1 big outer= 5 squaresAlways count the OUTER shape tooBig triangle split by 1 line → 2 small + 1 big = 3 triangles5-point star → 5 triangle points

Shapes hide inside other shapes. Count the smallest ones first, then the ones made by joining two or more of them, and then the whole outer shape. Add the lot together.

🪄 How to do it

  1. 1.Count the smallest triangles or squares in the picture above.
  2. 2.Look for larger ones made by joining those small pieces together.
  3. 3.Add the whole outer shape if it counts, then total everything up.

✍️ Try it

A is cut into 4 equal small triangles. How many triangles in total? Now three small squares sit in a row. How many squares in total?

The gives 4 small ones plus the whole outer one, which makes 5. The three squares in a row give just 3, because a row of three squares is a rectangle and not a bigger square.

⚠️ Watch out

Your first count is nearly always too low, because the medium shapes are the easy ones to miss. Before you answer, go back and look only for shapes built from two or three pieces joined together.

🌟 Why this matters

Seeing the bigger shape made of smaller ones is how you read a tiled wall or a chessboard. It is how you count the panes in a window without pointing at each one. Anyone laying tiles works the same way, in rows and blocks rather than one piece at a time.
Exam tip: Write your running total in three parts, small then medium then whole, because doing it in your head is how pieces get counted twice.

3

Watch me solve one

Quanta solves it

“Watch how I think through one of these — then you try.”

THE PROBLEM

“A big triangle is split into 4 small triangles. How many triangles in total?”

  1. 1

    Count the small shapes

    4 small triangles inside.

  2. 2

    Don't forget the BIG outer shape

    1 big triangle = the whole outer.

  3. 3

    Add them up

    4 + 1 = 5

  4. 4

    Answer ✓

    = 5 triangles

Now you try one — same steps, different story 👇

4

See it in action

⚡ Time to play!

Tap every small square, then tap the whole outline. Count each four-sided shape once.

COUNT · ROUND 1 OF 3

shapes counted: 0
5

Quick check

✅ VERIFICATION

A rectangle is cut into 2 equal smaller rectangles by one line. How many rectangles are in the figure in total?

⚡ Finish the activity above first to answer.

Keep going!

Up next: Cube Counting

Hidden cubes underneath still count.

Concept 7 of 9 →