Volume • Topic 2 of 3

Counting Cubic Units in Irregular 3D Shapes

Not every solid is a perfect box. Some shapes are made by stacking unit cubes in an uneven way (think of steps, an L-shape or a staircase of blocks). To find their volume we simply count every unit cube — including the ones hidden behind or underneath others.

There are three reliable ways to count, and they always give the same answer. Pick whichever suits the picture.

Method 1 — Layer by layer.

Count the cubes in the bottom layer.
Count the cubes in each layer above it.
Add all the layers together.

Method 2 — Height grid (top view). A top-view grid shows how many cubes are stacked in each column. Add all the column heights to get the total number of cubes.

Method 3 — Break it into boxes. Split the irregular shape into two or three rectangular prisms, find the volume of each with $V = l \times w \times h$, then add them.

The golden rule: hidden cubes count too. A cube that you cannot see because another cube sits in front of it is still part of the solid, so it must be counted.

Tip: after counting, you can often check your answer with a different method — if both methods agree, you can be confident it is right.

IRREGULAR SOLID: count EVERY unit cube, including hidden ones.

METHOD 1 - Layer by layer (add the cubes in each layer):

   Layer 1 (bottom) = 9 cubes
   Layer 2          = 4 cubes
   Layer 3 (top)    = 1 cube
   Total            = 9 + 4 + 1 = 14 cubes

METHOD 2 - Height grid (top view; each number = column height):

   +---+---+---+
   | 3 | 1 | 1 |
   +---+---+---+
   | 2 | 1 | 1 |   Total = 3+1+1 + 2+1+1 + 1+1+1 = 12 cubes
   +---+---+---+
   | 1 | 1 | 1 |
   +---+---+---+

METHOD 3 - Break into boxes:  Prism A (2x2x2)=8  +  Prism B (3x1x2)=6  =  14

Solved Examples

Worked Example

A height grid (top view) shows these column heights: row 1 = [2, 1, 1], row 2 = [1, 1, 1], row 3 = [1, 1, 1]. Find the volume.

Solution

Add every column height.
$2+1+1 + 1+1+1 + 1+1+1$
$= 2 + (8 \times 1) = 2 + 8 = 10$
Answer: $10$ cubic units

Worked Example

A solid has 15 cubes in the bottom layer, 8 in the middle layer and 3 in the top layer. Find its volume.

Solution

Layer method: add the layers.
$15 + 8 + 3 = 26$
Answer: $26$ cubic units

Worked Example

An L-shaped solid is made of two boxes: box A is $2 \times 2 \times 2$ and box B is $3 \times 1 \times 2$. Find the total volume.

Solution

Box A: $V = 2 \times 2 \times 2 = 8$
Box B: $V = 3 \times 1 \times 2 = 6$
Total $= 8 + 6 = 14$
Answer: $14$ cubic units

Worked Example

A staircase of cubes has 6 cubes on the bottom step, 4 on the next and 1 on top. How many unit cubes in all?

Solution

$6 + 4 + 1 = 11$
Answer: $11$ cubic units

Worked Example

A solid is built from a $3 \times 3$ bottom layer with a single $1 \times 1$ tower of 2 extra cubes on one corner. Find the volume.

Solution

Bottom layer: $3 \times 3 = 9$ cubes
Extra tower: $2$ cubes
Total $= 9 + 2 = 11$
Answer: $11$ cubic units

Key Points

For an irregular solid, count every unit cube — hidden cubes included.
Layer method: add the cubes in each layer from bottom to top.
Height-grid method: add all the column heights in the top view.
Decompose method: split into boxes, find each volume, then add.
Check your total with a second method whenever you can.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.When counting cubes in an irregular solid, hidden cubes:

Explanation: Every cube counts.

Q2.A solid has layers of $9$, $4$ and $1$ cubes. Its volume is:

Explanation: $9+4+1=14$.

Q3.In the height-grid method you add all the:

Explanation: Sum of column heights.

Q4.Two boxes of $8$ and $6$ cubes joined together have volume:

Explanation: $8+6=14$.

Practice more MCQs → 📝 Assignment · 35 marks