Volume of Compound 3D shapes Worksheet | Fun and Engaging PDF ... - Free Printable

Let's solve each of these compound 3D shape volume problems step by step.

We will break each shape into simpler rectangular prisms (cuboids), calculate the volume of each part, and then add or subtract as needed.

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🔷 Volume of a Cuboid:

$$
\text{Volume} = \text{length} \times \text{width} \times \text{height}
$$

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✔ Problem 1

This shape has two parts:

- Bottom part: A cuboid with dimensions:
$4\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm}$
Volume = $4 \times 2 \times 2 = 16\,\text{cm}^3$

- Top part: A cuboid on top with dimensions:
$2\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm}$
Volume = $2 \times 2 \times 2 = 8\,\text{cm}^3$

Total volume = $16 + 8 = \boxed{24\,\text{cm}^3}$

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✔ Problem 2

This is an "L" shape made of two cuboids.

- Left block:
Dimensions: $5\,\text{cm} \times 2\,\text{cm} \times 4\,\text{cm}$
Volume = $5 \times 2 \times 4 = 40\,\text{cm}^3$

- Right block:
Dimensions: $6\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm}$
Volume = $6 \times 2 \times 2 = 24\,\text{cm}^3$

Wait — but the total length is 6 cm. The left block is 5 cm long, so the right block must be only $6 - 5 = 1\,\text{cm}$ in depth? Let’s double-check.

Actually, from the diagram:
- The full base is 6 cm long.
- The height is 5 cm on the left and 2 cm on the right.
- Depth is 2 cm throughout.

So we can split it into:

- Front block (right side):
$6\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm} = 24\,\text{cm}^3$

- Back block (left side):
$5\,\text{cm} \times 2\,\text{cm} \times (5 - 2) = 5 \times 2 \times 3 = 30\,\text{cm}^3$

Wait — no. Better to think:

The bottom layer is $6 \times 2 \times 2 = 24\,\text{cm}^3$
On top of that, there’s an additional block of height 3 cm, width 5 cm, depth 2 cm:
$5 \times 2 \times 3 = 30\,\text{cm}^3$

But wait — this would give total $24 + 30 = 54$, but overlapping?

No — actually, the total height is 5 cm on the left and 2 cm on the right. So:

- Lower block: covers entire base: $6\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm} = 24\,\text{cm}^3$
- Upper block: sits on the left side, extending 5 cm in length, 2 cm depth, and height 3 cm (since 5 - 2 = 3):
$5 \times 2 \times 3 = 30\,\text{cm}^3$

✔ Total volume = $24 + 30 = \boxed{54\,\text{cm}^3}$

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✔ Problem 3

This is a three-level stepped shape.

Break it into three cuboids:

- Bottom layer: $5\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm} = 30\,\text{cm}^3$
- Middle layer: $3\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm} = 18\,\text{cm}^3$
- Top layer: $3\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm} = 18\,\text{cm}^3$

Wait — let's check dimensions carefully.

From diagram:
- Bottom: full base: $5\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm}$ → $5 \times 3 \times 2 = 30\,\text{cm}^3$
- Middle: centered on top, size $3\,\text{cm} \times 3\,\text{cm} \times 2\,\text{cm}$ → $18\,\text{cm}^3$
- Top: same size again: $3 \times 3 \times 2 = 18\,\text{cm}^3$

But wait — the height increases by 2 cm per level, and the top is 2 cm high, so yes.

Total volume = $30 + 18 + 18 = \boxed{66\,\text{cm}^3}$

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✔ Problem 4

This is a complex L-shaped prism.

Break it into three cuboids:

Look at it as:
- A bottom horizontal block
- A vertical block on the left
- A top horizontal block

Alternatively, use layering.

From bottom up:

#### Block 1: Bottom base
Dimensions: $4\,\text{cm} \times 4\,\text{cm} \times 2\,\text{cm}$ → $4 \times 4 \times 2 = 32\,\text{cm}^3$

#### Block 2: Middle vertical extension
On the left side, adds height from 2 cm to 4 cm, so extra height = 2 cm
Length = 3 cm, depth = 2 cm → $3 \times 2 \times 2 = 12\,\text{cm}^3$

Wait — depth is 2 cm? From diagram: the top block is 2 cm deep, and the whole structure has depth 4 cm?

Wait — look at labels:
- Right side: depth = 4 cm
- Top block: depth = 2 cm
- Left block: depth = 4 cm?

Actually, the depth (into page) is consistent at 4 cm for all parts.

Let’s re-analyze.

From the diagram:
- The entire object has depth = 4 cm
- The base is 3 cm wide and 4 cm deep → area = $3 \times 4 = 12\,\text{cm}^2$
- Height varies.

Better to divide into three parts:

1. Left front block:
Height: 4 cm, width: 3 cm, depth: 4 cm → $3 \times 4 \times 4 = 48\,\text{cm}^3$

2. Right back block:
Width: 1 cm, depth: 4 cm, height: 2 cm → $1 \times 4 \times 2 = 8\,\text{cm}^3$

3. Top middle block:
Width: 2 cm, depth: 2 cm, height: 1 cm → $2 \times 2 \times 1 = 4\,\text{cm}^3$

Wait — not matching.

Alternative method: split into two main parts.

Let’s try:

- Part A: The large base block:
Size: $4\,\text{cm} \times 4\,\text{cm} \times 2\,\text{cm}$ → $4 \times 4 \times 2 = 32\,\text{cm}^3$

- Part B: The vertical extension on the left:
It rises 2 cm more (from 2 to 4), width = 3 cm, depth = 4 cm → $3 \times 4 \times 2 = 24\,\text{cm}^3$

- Part C: The small top block on the right:
Size: $2\,\text{cm} \times 2\,\text{cm} \times 1\,\text{cm}$ → $2 \times 2 \times 1 = 4\,\text{cm}^3$

Wait — is this overlapping?

Actually, better to see the cross-section.

Let me go back.

Looking at the shape:

It looks like:
- A rectangular prism of $4 \times 4 \times 2$ = 32 cm³ (bottom)
- On top of that, a smaller prism of $2 \times 2 \times 1$ = 4 cm³ (on the right)
- And on the left, a tall prism of $3 \times 4 \times 2$ = 24 cm³ (but already included?)

Wait — confusion.

Let’s use horizontal layers.

Layer 1 (bottom): height = 2 cm, base = 4 cm × 4 cm → $4 \times 4 \times 2 = 32\,\text{cm}^3$

Layer 2 (middle): height = 2 cm, but only on left side:
- Left part: width = 3 cm, depth = 4 cm → $3 \times 4 \times 2 = 24\,\text{cm}^3$

Layer 3 (top): height = 1 cm, only on right side:
- Width = 2 cm, depth = 2 cm → $2 \times 2 \times 1 = 4\,\text{cm}^3$

Wait — but the top block is only 2 cm deep, while the rest is 4 cm deep?

Yes! So:

- Bottom layer: full base: $4\,\text{cm} \times 4\,\text{cm} \times 2\,\text{cm} = 32\,\text{cm}^3$
- Middle layer: only on left: $3\,\text{cm} \times 4\,\text{cm} \times 2\,\text{cm} = 24\,\text{cm}^3$
- Top layer: only on right: $2\,\text{cm} \times 2\,\text{cm} \times 1\,\text{cm} = 4\,\text{cm}^3$

Wait — but the middle layer is already 2 cm tall, and the top is 1 cm, so total height is 5 cm?

No — the labels say:
- Right side: height = 4 cm
- Left side: height = 4 cm
- But the top block is only 1 cm high

Wait — re-examining:

From the diagram:
- The total height on the left is 4 cm
- On the right, the base is 2 cm, then a 1 cm top block → total 3 cm?
But label says "4 cm" on the right side.

Ah — perhaps the right side is 4 cm tall, and the top block is 1 cm, so the base is 3 cm?

But label shows "4 cm" on the right side.

Wait — the label says:
- Right side: 4 cm (height)
- Top block: 1 cm high
- So base must be 3 cm high?

But earlier we have 2 cm base?

I think I misread.

Let’s re-analyze the diagram:

- The bottom is 2 cm high
- Then a middle layer of 2 cm height on the left
- Then a top layer of 1 cm on the right

But total height on the right is 2 + 1 = 3 cm? But labeled as 4 cm.

Wait — maybe the depth is what's being shown.

Let’s read the labels:

- Depth: 4 cm (right side)
- Height: 4 cm (right side)
- Top block: 1 cm high, 2 cm wide, 2 cm deep
- Base: 3 cm wide, 4 cm deep, 2 cm high
- Left extension: 3 cm wide, 4 cm deep, 2 cm high?

Wait — perhaps it's better to break into two blocks:

1. Main block: $4\,\text{cm} \times 4\,\text{cm} \times 2\,\text{cm} = 32\,\text{cm}^3$

2. Left extension: extends upward by 2 cm, width = 3 cm, depth = 4 cm → $3 \times 4 \times 2 = 24\,\text{cm}^3$

3. Right top block: $2\,\text{cm} \times 2\,\text{cm} \times 1\,\text{cm} = 4\,\text{cm}^3$

But now total = $32 + 24 + 4 = 60\,\text{cm}^3$

But is there overlap?

No — the left extension is on the left, the top block is on the right.

But the main block is $4 \times 4 \times 2$ — so it covers the whole base.

Then the left extension adds another 2 cm height on the left side: $3 \times 4 \times 2 = 24$

Then the right top block is on the right, but only 2 cm deep and 2 cm wide, so it's sitting on the main block.

But the main block is 4 cm deep, so the top block is only 2 cm deep — so it's partially over the main block.

So total volume:

- Main block: $4 \times 4 \times 2 = 32$
- Left extension: $3 \times 4 \times 2 = 24$
- Right top block: $2 \times 2 \times 1 = 4$

But are they all separate?

Wait — the left extension is on top of the main block, and the right top block is also on top.

But the left extension is 3 cm wide, 4 cm deep, 2 cm high — so it covers the entire left part.

The right top block is 2 cm wide, 2 cm deep, 1 cm high — so it's on the right, but only half the depth.

So total volume = $32 + 24 + 4 = \boxed{60\,\text{cm}^3}$

Yes, because no overlap between the added parts.

✔ Answer: 60 cm³

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✔ Problem 5

This is an "L" shape made of two cuboids.

Break into two parts:

- Vertical block:
Dimensions: $2\,\text{cm} \times 5\,\text{cm} \times 5\,\text{cm}$ → $2 \times 5 \times 5 = 50\,\text{cm}^3$

- Horizontal block:
Dimensions: $7\,\text{cm} \times 1\,\text{cm} \times 3\,\text{cm}$ → $7 \times 1 \times 3 = 21\,\text{cm}^3$

But wait — do they overlap?

Yes — they connect at the corner.

The common part is $5\,\text{cm} \times 1\,\text{cm} \times 3\,\text{cm}$ — but actually, the horizontal block is only 1 cm high, and the vertical block is 5 cm high, so the overlap is only in the shared region.

But since both blocks are solid, we must subtract the overlapping volume if we add them.

But in this case, the horizontal block is attached to the bottom of the vertical block, and the vertical block is 5 cm high, horizontal is 1 cm high, so they share a face.

But since it's a solid, the volume is just the sum, as long as we don't double-count.

But actually, the horizontal block is not under the vertical block — it's attached to the side.

Wait — looking at the diagram:

- Vertical block: 5 cm wide, 5 cm deep, 2 cm high
- Horizontal block: 7 cm long, 3 cm wide, 1 cm high

They meet at the corner.

But the horizontal block is only 1 cm high, and the vertical block is 2 cm high — so they do not overlap in height.

But the horizontal block extends from the base, and the vertical block is on top of it?

Wait — no. The horizontal block is 1 cm high, and the vertical block is 2 cm high, but they are connected at the base.

But the horizontal block is 3 cm wide, and the vertical block is 5 cm wide.

So the overlap is $5\,\text{cm} \times 3\,\text{cm} \times 1\,\text{cm}$ — but wait, the horizontal block is only 1 cm high, and the vertical block is 2 cm high, so the shared base is $5\,\text{cm} \times 3\,\text{cm}$, but the horizontal block is only 1 cm high.

But the vertical block sits on the same base as the horizontal block?

No — the vertical block is 5 cm wide, 5 cm deep, 2 cm high — so its base is $5 \times 5$

The horizontal block is 7 cm long, 3 cm wide, 1 cm high — so its base is $7 \times 3$

They intersect at $5 \times 3$ — but since both are on the ground, and we're adding volumes, we need to avoid double-counting.

But the vertical block is 5 cm deep, the horizontal block is only 3 cm wide — so they share a 5×3 area on the base.

But since both are attached to the same base, and we’re calculating volume, we can compute:

- Vertical block: $5 \times 5 \times 2 = 50\,\text{cm}^3$
- Horizontal block: $7 \times 3 \times 1 = 21\,\text{cm}^3$

But the overlapping region is $5 \times 3 \times 1 = 15\,\text{cm}^3$ — but this region is only counted once in the total, because it's the base.

Wait — no: the vertical block is 2 cm high, so it occupies $5 \times 5 \times 2 = 50$

The horizontal block is $7 \times 3 \times 1 = 21$, but only 5 cm of its length overlaps with the vertical block's base.

But since both are on the ground, and the horizontal block is in front of the vertical block, they might not overlap in space.

Looking at the diagram: the horizontal block extends forward, and the vertical block is behind it.

But the vertical block is 5 cm deep, and the horizontal block is 3 cm wide — likely, the horizontal block is attached to the side of the vertical block.

But the vertical block is 5 cm wide, 5 cm deep, 2 cm high.

The horizontal block is 7 cm long, 3 cm wide, 1 cm high — so it’s only 1 cm high, so it doesn’t reach the top.

They are joined at the base, but their bases overlap in a rectangle of $5 \times 3$ cm².

But since both are solid, and the horizontal block is on the ground, and the vertical block is also on the ground, the overlap is only in the base, but not in volume — they are separate solids touching.

So total volume = $50 + 21 = \boxed{71\,\text{cm}^3}$

But wait — is the horizontal block under the vertical block? No, because the vertical block is 2 cm high, and the horizontal block is only 1 cm high.

But the horizontal block is attached to the side — so they are adjacent.

Yes, so no volume overlap.

✔ Answer: 71 cm³

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✔ Problem 6

This is a large cuboid with a smaller cuboid removed.

- Outer cuboid: $8\,\text{cm} \times 6\,\text{cm} \times 4\,\text{cm}$ → $8 \times 6 \times 4 = 192\,\text{cm}^3$

- Inner hollow: a cuboid of $4\,\text{cm} \times 2\,\text{cm} \times 2\,\text{cm}$ → $4 \times 2 \times 2 = 16\,\text{cm}^3$

But is it completely removed?

Yes — it's a hole inside.

So volume = outer – inner = $192 - 16 = \boxed{176\,\text{cm}^3}$

Wait — is the inner cuboid fully within the outer one?

Outer: 8×6×4
Inner: 4×2×2 — yes, fits.

And the diagram shows it’s a cavity.

✔ Answer: 176 cm³

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✔ Final Answers:

1) $ \boxed{24\,\text{cm}^3} $
2) $ \boxed{54\,\text{cm}^3} $
3) $ \boxed{66\,\text{cm}^3} $
4) $ \boxed{60\,\text{cm}^3} $
5) $ \boxed{71\,\text{cm}^3} $
6) $ \boxed{176\,\text{cm}^3} $

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Let me know if you'd like diagrams or further explanation!