Concept 3 of 9
Embedded Figures: Count All the Shapes
Today's big idea
Count the smallest shapes first, then the ones built from two, then from three.
Count the smallest pieces first, then every shape made by joining them.
How it works
👀 What's going on?
A figure looks like four triangles until somebody points out the bigger ones made from two of them, and your count jumps.
💡 Here's the trick
Look at the figure above. Inner lines make shapes at more than one size. Counting the smallest first, then the ones built from two, then from three, keeps the tally honest.
🪄 How to do it
- 1.Count the smallest shapes in the figure above.
- 2.Count the shapes made by joining two of those.
- 3.Keep going up in size, then add every group together.
✍️ Try it
A square with both diagonals drawn holds how many triangles? Then: how many squares of any size are in a 3 by 3 grid?
The diagonals make 4 small triangles, plus 4 larger ones each made of two halves, giving 8. The grid has 9 small, 4 of size two and 1 of size three, giving 14.
⚠️ Watch out
You might be thinking the small shapes are all of them. A square with both diagonals holds 4 small triangles. Each half of the square is a triangle too, so the answer is 8.
🌟 Why this matters
Reading a floor plan or a tiling pattern means seeing the big shapes as well as the small ones. A builder who counts only the small squares orders the wrong amount of tile. That same undercount is the one this topic keeps testing.
Exam tip: Write your count size by size.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“How many squares are in a 3×3 grid of equal cells?”
- 1
Smallest first
9 individual 1×1 cells.
- 2
Combine into 2×2
4 different 2×2 squares fit inside.
- 3
The outer 3×3
+ 1 (the whole outline).
- 4
Total
9 + 4 + 1 = 14
- 5
Answer ✓
14
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap each triangle hidden inside the figure — including the big ones. Find them all.
HIDDEN SHAPES · ROUND 1 OF 3
Figure: a square with both diagonals
Quick check
✅ VERIFICATION
A square is divided by ONE diagonal only. How many triangles are there in total?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Figure Matrix
Sum, product, or arithmetic, test row-wise then column-wise.