Volume of Compound 3D Shapes worksheet with six geometric figures and dimensions for calculating volume.

Let's solve each of these compound 3D shapes step by step. The goal is to find the volume of each shape.

We’ll use the formula for the volume of a rectangular prism (cuboid):

> Volume = Length × Width × Height

For compound shapes, we break them down into simpler rectangular prisms, calculate the volume of each, and then add or subtract as needed.

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🔹 Problem 1)

This shape consists of two rectangular blocks:

- Bottom block:
- Length = 4 cm
- Width = 2 cm
- Height = 2 cm
→ Volume = 4 × 2 × 2 = 16 cm³

- Top block:
- Length = 2 cm
- Width = 2 cm
- Height = 2 cm
→ Volume = 2 × 2 × 2 = 8 cm³

✔ Total volume = 16 + 8 = 24 cm³

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🔹 Problem 2)

This shape can be split into two parts:

- Front block (left):
- Length = 5 cm
- Width = 2 cm
- Height = 6 cm
→ Volume = 5 × 2 × 6 = 60 cm³

- Back block (right):
- It’s on top of the front one but only extends 4 cm in length.
- Length = 4 cm
- Width = 2 cm
- Height = 6 cm
→ Volume = 4 × 2 × 6 = 48 cm³

Wait — actually, looking closely, the height of the back block is only 4 cm, since it sits on top of the lower block.

But wait — the total height is 5 cm at the left, and 4 cm at the right? Let's re-analyze.

Actually, the entire shape has depth of 6 cm, and width of 2 cm.

Looking at the side view:
- Left part: height = 5 cm
- Right part: height = 4 cm

So it's better to split it into two horizontal blocks:

- Lower block (full base):
- Length = 6 cm
- Width = 2 cm
- Height = 4 cm
→ Volume = 6 × 2 × 4 = 48 cm³

- Upper block (on left):
- Length = 2 cm (since the height difference is only on the left)
- Width = 2 cm
- Height = 1 cm (5 − 4 = 1)
→ Volume = 2 × 2 × 1 = 4 cm³

✔ Total volume = 48 + 4 = 52 cm³

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🔹 Problem 3)

Split into three blocks:

- Bottom block (base):
- Length = 5 cm
- Width = 3 cm
- Height = 2 cm
→ Volume = 5 × 3 × 2 = 30 cm³

- Middle block (center):
- Length = 3 cm
- Width = 3 cm
- Height = 2 cm
→ Volume = 3 × 3 × 2 = 18 cm³

- Top block (top-left):
- Length = 3 cm
- Width = 3 cm
- Height = 2 cm
→ Wait, no — this is already included?

Wait — actually, the middle block is raised above the base. But let’s look carefully.

Actually, the entire structure has:
- A base layer of 5 cm long, 3 cm wide, 2 cm high → 5×3×2 = 30 cm³
- On top of that, there's a central block of 3 cm long, 3 cm wide, 2 cm high → 3×3×2 = 18 cm³
- But wait — the top block is only 2 cm high, so if the base is already 2 cm, this would be 4 cm total height? No — the drawing shows the middle block is 2 cm tall, and the base is also 2 cm.

But the total height is 2 cm + 2 cm = 4 cm? No — the labels show 2 cm on the side, and the top block is 2 cm high, so likely the entire height is 2 cm for base and 2 cm for top.

Wait — the label says 2 cm on the side, and the total height is 2 cm? That doesn't make sense.

Let me re-express:

From the diagram:
- The bottom is 2 cm high.
- The middle block is 2 cm high — so total height is 4 cm?
But the label says "2 cm" next to the vertical edge.

Wait — actually, the vertical dimension shown is 2 cm for the base, and the top block adds another 2 cm?

No — the side label says 2 cm for the bottom, and the top block is labeled 2 cm — but it's stacked on top.

Wait — perhaps the entire height is 2 cm, and the depth is 3 cm?

Let’s reinterpret.

Actually, the label on the side is 2 cm, and the top block has 2 cm height, but it’s not stacked — it's level with the base?

No — from the diagram, the middle block is raised.

Wait — the height of the entire object is 2 cm at the base, and 2 cm more on top — so total height = 4 cm?

But the label says 2 cm next to the base — maybe that's the height of the base, and the top block is also 2 cm high, so total height is 4 cm?

Yes — likely.

So:

- Base block:
- Length = 5 cm
- Width = 3 cm
- Height = 2 cm
→ Volume = 5 × 3 × 2 = 30 cm³

- Middle block:
- Length = 3 cm
- Width = 3 cm
- Height = 2 cm
→ Volume = 3 × 3 × 2 = 18 cm³

- Top block:
- The top block is 3 cm long, 3 cm wide, 2 cm high — but wait, is it?

Wait — the top block is 3 cm long, 3 cm wide, 2 cm high — but it’s placed on top of the middle block?

But the middle block is already 2 cm high, so adding another 2 cm makes it 4 cm?

But the side label says 2 cm — so perhaps the height of the entire shape is 2 cm?

Wait — I think I'm misreading.

Let’s go back.

The label on the left side says 2 cm, and the top block is labeled 2 cm — but it’s not taller than the base.

Wait — the height of the entire object is 2 cm? Then why is the top block labeled 2 cm?

No — the vertical dimension is 2 cm, meaning the height of the base is 2 cm.

But the top block is also 2 cm high — but it’s stacked, so total height is 4 cm?

But the label says 2 cm next to the base — probably the height of the base is 2 cm, and the top block is 2 cm high, so total height is 4 cm.

But the width of the top block is 3 cm, and length is 3 cm.

Wait — the top block is 3 cm long, 3 cm wide, 2 cm high — but it's on top of the middle block, which is 3 cm long, 3 cm wide, 2 cm high.

But the base is 5 cm long, 3 cm wide, 2 cm high.

So:

- Base: 5 × 3 × 2 = 30 cm³
- Middle block: 3 × 3 × 2 = 18 cm³
- Top block: 3 × 3 × 2 = 18 cm³

But wait — the top block is not on top of the middle block — it's next to it?

No — the diagram shows a step-like shape.

Actually, the middle block is on top of the base, and the top block is on top of the middle block?

But the labels say:

- Base: 5 cm (length), 3 cm (width), 2 cm (height)
- Middle block: 3 cm (length), 3 cm (width), 2 cm (height)
- Top block: 3 cm (length), 3 cm (width), 2 cm (height)

But the total height would be 2 + 2 + 2 = 6 cm? That seems too much.

Wait — no. The vertical label on the side says 2 cm, and the top block is labeled 2 cm — but it’s not taller.

Actually, the height of the entire shape is 2 cm, and all blocks are 2 cm high — so they’re all at the same level?

But that can’t be — the top block is above the base.

I think the label “2 cm” on the side refers to the height of the base, and the top block is also 2 cm high, so total height is 4 cm.

But the label says 2 cm — perhaps it’s the depth?

Wait — let’s check the dimensions:

- Bottom: 5 cm (length), 3 cm (width), 2 cm (height)
- Middle block: 3 cm (length), 3 cm (width), 2 cm (height)
- Top block: 3 cm (length), 3 cm (width), 2 cm (height)

But the top block is placed on top of the middle block, so its height is 2 cm, but the middle block is already 2 cm high — so total height is 4 cm.

But the side label says 2 cm — that must be the height of the base, not the total.

So yes.

So:

- Base: 5 × 3 × 2 = 30 cm³
- Middle block: 3 × 3 × 2 = 18 cm³
- Top block: 3 × 3 × 2 = 18 cm³

But wait — the middle block is on top of the base, so it's not overlapping?

Actually, the middle block is on top of the base, and the top block is on top of the middle block.

But the base is 5 cm long, and the middle block is 3 cm long — so it fits.

But the top block is 3 cm long — so it's on top of the middle block.

So total volume = 30 + 18 + 18 = 66 cm³

But wait — the width is 3 cm for all, and depth is 3 cm.

But the base has length 5 cm, so it’s 5 cm long, 3 cm deep, 2 cm high.

The middle block is 3 cm long, 3 cm deep, 2 cm high — placed on top of the base.

The top block is 3 cm long, 3 cm deep, 2 cm high — placed on top of the middle block.

So yes.

✔ Total volume = 30 + 18 + 18 = 66 cm³

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Wait — but the top block is not extending beyond — it's centered?

But the diagram shows a step-like shape, so likely:

- Base: 5 × 3 × 2 = 30
- Middle block: 3 × 3 × 2 = 18
- Top block: 3 × 3 × 2 = 18

But is the top block really 3 cm long? Yes.

But the total length is 5 cm, so the middle block is 3 cm, and the top block is 3 cm — but they might overlap?

No — the top block is on top of the middle block, so it’s fine.

But the middle block is on top of the base, so it’s not double-counting.

So ✔ 66 cm³

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🔹 Problem 4)

This is a zigzag shape.

Break it into three blocks:

- Left block:
- Length = 3 cm
- Width = 4 cm
- Height = 2 cm
→ Volume = 3 × 4 × 2 = 24 cm³

- Middle block:
- Length = 4 cm
- Width = 4 cm
- Height = 2 cm
→ Volume = 4 × 4 × 2 = 32 cm³

- Right block:
- Length = 2 cm
- Width = 4 cm
- Height = 1 cm
→ Volume = 2 × 4 × 1 = 8 cm³

Total volume = 24 + 32 + 8 = 64 cm³

But wait — the height of the right block is 1 cm, and the middle block is 2 cm high — so the right block is shorter.

But the label says 1 cm on the top of the right block.

Yes — so the right block is only 1 cm high.

And the middle block is 2 cm high.

And the left block is 2 cm high.

So:

- Left: 3 × 4 × 2 = 24
- Middle: 4 × 4 × 2 = 32
- Right: 2 × 4 × 1 = 8

✔ Total = 64 cm³

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🔹 Problem 5)

This is an L-shaped block.

Break into two parts:

- Vertical block (left):
- Length = 5 cm
- Width = 2 cm
- Height = 5 cm
→ Volume = 5 × 2 × 5 = 50 cm³

- Horizontal block (right):
- Length = 7 cm
- Width = 1 cm
- Height = 3 cm
→ Volume = 7 × 1 × 3 = 21 cm³

But wait — do they overlap?

The horizontal block starts at the end of the vertical block.

The vertical block is 5 cm high, and the horizontal block is 3 cm high — so they meet at the base.

But the width of the vertical block is 2 cm, and the horizontal block is 1 cm — so they don't overlap in width.

But the horizontal block is attached to the side of the vertical block.

So:

- Vertical block: 5 × 2 × 5 = 50 cm³
- Horizontal block: 7 × 1 × 3 = 21 cm³

But the horizontal block is 3 cm high, and the vertical block is 5 cm high, so they are joined at the bottom.

But the horizontal block has width 1 cm, and the vertical block has width 2 cm — so they are adjacent.

So no overlap.

✔ Total volume = 50 + 21 = 71 cm³

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🔹 Problem 6)

This is a large box with a smaller box removed.

- Outer box:
- Length = 8 cm
- Width = 4 cm
- Height = 6 cm
→ Volume = 8 × 4 × 6 = 192 cm³

- Inner hole:
- It's a smaller box inside:
- Length = 4 cm
- Width = 2 cm
- Height = 4 cm
→ Volume = 4 × 2 × 4 = 32 cm³

So total volume = Outer – Inner = 192 – 32 = 160 cm³

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✔ Final Answers:

1) 24 cm³
2) 52 cm³
3) 66 cm³
4) 64 cm³
5) 71 cm³
6) 160 cm³

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📝 Summary of Solutions:

| Problem | Volume |
|--------|--------|
| 1 | 24 cm³ |
| 2 | 52 cm³ |
| 3 | 66 cm³ |
| 4 | 64 cm³ |
| 5 | 71 cm³ |
| 6 | 160 cm³ |

Let me know if you'd like a visual breakdown of any shape!