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Concept 4 of 8

Counting Figures: Triangles & Squares

1

Today's big idea

Count by size, smallest pieces first, and remember the whole outline is a shape too.

Pieces joined in twos and threes make shapes that a first count always misses.

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How it works

👀 What's going on?

You are shown one figure with a few lines through it and asked how many triangles it contains. The first count is almost always too low, because the big shapes hide behind the small ones.

💡 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

Look at the figure above. Counting the smallest pieces is only the start. Two small pieces side by side often make a larger one, and the whole outline is a shape too. Count by size, smallest first, and add the groups at the end.

🪄 How to do it

  1. 1.Count the smallest pieces in the figure above and write that number down.
  2. 2.Look for shapes made of two or more pieces joined together.
  3. 3.Add the outer shape last, then total the lot.

✍️ Try it

How many rectangles of all sizes are in a rectangle cut into 3 equal strips? Then: how many squares of all sizes are in a 2 by 2 grid?

The strips give 3 single ones, 2 made from neighbouring pairs and 1 whole, making 6. The grid gives 4 small squares and the 1 big one, making 5.

⚠️ Watch out

You might be thinking the answer is just the number of pieces you can see. Cut a rectangle into three strips and the obvious answer is three. But two strips together make a rectangle, and so do all three. The count is nearly always bigger than it first looks.

🌟 Why this matters

Careful counting without missing or repeating anything is a habit worth more than any single trick. Tiling, packing and stacking all reward the same patience.
Exam tip: Write a small tally for each size rather than one running total, since that is what keeps the big shapes from vanishing.

3

Watch me solve one

Quanta solves it

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

THE PROBLEM

“In a 2-by-2 grid of squares, how many rectangles of all sizes are there?”

  1. 1

    Count the small squares

    4 small squares fill the grid.

  2. 2

    Add the wider and taller rectangles

    Each row and each column gives a 1×2 rectangle: that is 4 more.

  3. 3

    Add the whole outer square

    4 + 4 + 1 = 9

  4. 4

    Answer ✓

    = 9 rectangles

Now you try one — same steps, different story 👇

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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

When counting shapes inside a figure, the most commonly missed one is

⚡ Finish the activity above first to answer.

Keep going!

Up next: Figure Matrix

Say the rule out loud before you pick.

Concept 5 of 8 →