Concept 6 of 8
Cube Stack: Count 3D Cubes
Today's big idea
Count the cubes layer by layer and add, or multiply the three sides of a solid block.
Hidden cubes count too, and in a big block they outnumber the ones you see.
How it works
👀 What's going on?
A picture of stacked cubes is flat, but the thing it shows is not. Some cubes hide behind or underneath the ones you can see. Counting only the visible faces gives the wrong answer every time.
💡 Here's the trick
Look at the stack above. Taking it one layer at a time turns a solid into a few easy counts that you then add. For a solid block there is a quicker route still, since length times width gives one layer and multiplying by the height gives the lot.
🪄 How to do it
- 1.Find the bottom layer of the stack above and count the cubes in it.
- 2.Count each layer going upwards, writing each number down.
- 3.Add the layers together.
✍️ Try it
A stack has 5 cubes on the bottom, 3 in the middle and 1 on top. Then: a solid block is 3 cubes long, 2 wide and 2 tall.
The layers give 5 plus 3 plus 1, which is 9 cubes. The block has 3 times 2, which is 6 in one layer, and two layers make 12 cubes.
⚠️ Watch out
You might be thinking you can count the cubes you can see. In a solid block the middle cubes never show at all, and in bigger blocks the hidden ones outnumber the visible. Count layers, not faces.
🌟 Why this matters
Working out how many boxes fit in a crate, or how many bricks are in a wall, is this same layer count. It is also the first step towards volume.
Exam tip: If the shape is a plain block, multiply the three side lengths rather than counting anything.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A staircase stack has bottom 4 cubes, middle 2, top 1. How many cubes?”
- 1
Count each layer
Bottom = 4. Middle = 2. Top = 1.
- 2
Add all layers
4 + 2 + 1 = 7
- 3
Answer ✓
= 7 cubes
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap each layer to count its cubes. Add them all up for the total.
STACK · ROUND 1 OF 3
Quick check
✅ VERIFICATION
In a picture of stacked cubes, the cubes you cannot see are
⚡ Finish the activity above first to answer.
Keep going!
Up next: Dice
1-6, 2-5, 3-4. Bottom = 7 − top.