Math worksheet for calculating the volume of compound solids.

Let's solve each compound solid step by step. The key idea is to break each compound solid into simpler rectangular prisms (cuboids), calculate the volume of each part, and then add them together.

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🔷 Volume of a Rectangular Prism:

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

We'll use this formula for each part.

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🔹 1st Solid (Top Left)

This shape consists of two rectangular prisms:

- Bottom part:
- Length = 5 cm
- Width = 4 cm
- Height = 2 cm
$$
V_1 = 5 \times 4 \times 2 = 40 \text{ cm}^3
$$

- Top part:
- Length = 4 cm
- Width = 4 cm
- Height = 5 cm (but only 3 cm above the bottom?) Wait — let’s analyze carefully.

Actually, looking at the diagram:
- The top block is 4 cm × 4 cm × 5 cm? But it's sitting on top of the lower one.
Wait — the total height is 5 cm, and the bottom block has height 2 cm, so the top block must be 3 cm high?

But wait — the top block is drawn as 4 cm in depth, 4 cm in width, and height appears to be from the base up to 5 cm total.

Looking again: the entire solid has a height of 5 cm, and the lower part is 2 cm high, so the upper part is 3 cm high.

So:
- Lower block: 5 cm (length) × 4 cm (width) × 2 cm (height) = $5 \times 4 \times 2 = 40$ cm³
- Upper block: 4 cm (length) × 4 cm (width) × 3 cm (height) = $4 \times 4 \times 3 = 48$ cm³

✔ Total Volume = $40 + 48 = \boxed{88} \text{ cm}^3$

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🔹 2nd Solid (Top Right)

This looks like two rectangles side by side.

- Left prism:
- Dimensions: 4 cm (length) × 4 cm (width) × 8 cm (height)
$$
V_1 = 4 \times 4 \times 8 = 128 \text{ cm}^3
$$

- Right prism:
- Dimensions: 5 cm (length) × 2 cm (width) × 8 cm (height)
$$
V_2 = 5 \times 2 \times 8 = 80 \text{ cm}^3
$$

✔ Total Volume = $128 + 80 = \boxed{208} \text{ cm}^3$

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🔹 3rd Solid (Middle Left)

This is an L-shaped solid made of two parts.

Break it into two rectangles:

- Left block:
- Length = 7 cm
- Width = 4 cm
- Height = 8 cm
$$
V_1 = 7 \times 4 \times 8 = 224 \text{ cm}^3
$$

- Right block:
- Length = 5 cm
- Width = 4 cm
- Height = 9 cm? Wait — no! The height is consistent with the left block? No.

Wait — the height of the entire structure is 8 cm, but the right block extends only 4 cm in height? Let’s look carefully.

Wait — actually, both blocks have same height? No — the left block is 8 cm tall, and the right block is also 8 cm tall? But the drawing shows the right block is attached, and its height is not labeled.

But the left block has height 8 cm, and the right block has length 5 cm, width 4 cm, and height must be 8 cm too? Wait — no.

Wait: the horizontal dimension is labeled: left part is 7 cm long, right part is 5 cm long, and the vertical height is 8 cm for the whole thing. But the depth is 4 cm for both.

But the height is the same — 8 cm — for both.

So:

- Left block: 7 cm × 4 cm × 8 cm = $7 \times 4 \times 8 = 224$ cm³
- Right block: 5 cm × 4 cm × 8 cm = $5 \times 4 \times 8 = 160$ cm³

✔ Total Volume = $224 + 160 = \boxed{384} \text{ cm}^3$

Wait — but the right block is not aligned in height? Actually, yes — it seems both are full height.

But wait — the drawing shows the right block is shorter in height? No — the vertical line goes up to 8 cm, and the right block is attached to the top of the left block? No — it's attached horizontally.

Actually, the entire structure has a height of 8 cm, and both parts are full height.

Yes, both are 8 cm high, so:

✔ Total Volume = $224 + 160 = \boxed{384} \text{ cm}^3$

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🔹 4th Solid (Middle Right)

This is a stack of two blocks.

- Bottom block:
- Dimensions: 5 cm (length) × 6 cm (width) × 7 cm (height)? Wait — let’s read labels.

Labeling:
- Bottom block: 5 cm (length), 6 cm (width), and height is 6 cm? Wait — the height of the bottom block is 6 cm?
But the top block is 3 cm high, and total height is 7 cm.

Wait — the total height is 7 cm, and the top block is 3 cm high → so bottom block is 4 cm high?

No — the bottom block has height 6 cm, and the top block is 3 cm high? But that would make total height 9 cm — but label says 7 cm.

Wait — recheck.

From diagram:
- The bottom block has:
- Length = 5 cm
- Width = 6 cm
- Height = 6 cm? But the top block is placed on top, and its height is 3 cm, and total height is 7 cm → so bottom block must be 4 cm high?

But the label says 7 cm on the side — that’s the total height.

So:
- Bottom block: height = 7 cm? But the top block is 3 cm high, so bottom block must be 4 cm high? But it's labeled 6 cm?

Wait — let’s read carefully.

The bottom block has:
- Length = 5 cm
- Width = 6 cm
- Height = 6 cm? But then the top block is 3 cm high → total height = 6 + 3 = 9 cm, but the side label says 7 cm.

That can't be.

Wait — perhaps the bottom block is only 4 cm high, and the top block is 3 cm high, making total height 7 cm.

But the label on the side says “7 cm” — which is the total height.

Now, the bottom block has:
- Depth = 6 cm (width)
- Length = 5 cm
- Height = ? → must be 4 cm if top is 3 cm and total is 7 cm.

But the top block is shown to be 3 cm high, and it sits on the bottom block.

But the bottom block is only 4 cm high, and the top block is 3 cm high → total 7 cm.

Yes.

So:
- Bottom block: 5 cm × 6 cm × 4 cm = $5 \times 6 \times 4 = 120$ cm³
- Top block: 3 cm × 4 cm × 3 cm = $3 \times 4 \times 3 = 36$ cm³

✔ Total Volume = $120 + 36 = \boxed{156} \text{ cm}^3$

Wait — but the top block has width 4 cm? And it's placed on the front face of the bottom block?

Yes — the top block is smaller: 3 cm (length) × 4 cm (width) × 3 cm (height)

And it sits on the bottom block, which is 5 cm × 6 cm × 4 cm.

So yes.

✔ Volume = $120 + 36 = \boxed{156} \text{ cm}^3$

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🔹 5th Solid (Bottom Left)

This is a stepped shape.

Break into two blocks:

- Bottom block:
- Length = 5 cm
- Width = 5 cm
- Height = 4 cm
$$
V_1 = 5 \times 5 \times 4 = 100 \text{ cm}^3
$$

- Top block:
- Length = 5 cm
- Width = 5 cm
- Height = 2 cm
$$
V_2 = 5 \times 5 \times 2 = 50 \text{ cm}^3
$$

Wait — but the top block is only 2 cm high, and the bottom block is 4 cm high? But the total height is 6 cm? Yes — 4 + 2 = 6 cm.

But the top block is centered? Yes — it's directly on top.

So total volume = $100 + 50 = \boxed{150} \text{ cm}^3$

✔ Volume = 150 cm³

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🔹 6th Solid (Bottom Right)

This is a solid with a smaller block removed or attached?

Wait — it looks like a large cube with a smaller block on the side.

But the labels:

- Large block: 6 cm × 5 cm × 6 cm? Wait — dimensions:
- Front: 6 cm (length), 5 cm (height), 6 cm (width)? Wait.

Let’s read:

- Base: 6 cm (length), 6 cm (width), height = 5 cm? But there's a top block.

Wait — the large block has:
- Length = 6 cm
- Width = 6 cm
- Height = 5 cm

Then a smaller block is attached on the side:
- It has: 3 cm (length), 5 cm (height), 6 cm (width)? No — the depth is 6 cm?

Wait — the smaller block is attached to the side and has:
- Length = 3 cm
- Height = 5 cm
- Width = 6 cm? But that can’t be — because the large block is 6 cm wide.

Wait — actually, the smaller block is in front of the large block, extending out.

But the large block is 6 cm wide (depth), and the smaller block is 3 cm wide? Or vice versa?

Wait — the diagram shows:

- The main block: 6 cm (length), 6 cm (width), 5 cm (height)
- A smaller block attached to the side, 3 cm deep, 5 cm high, and 5 cm long?

Wait — the smaller block has:
- Length = 5 cm
- Width = 3 cm
- Height = 5 cm

And it's attached to the side of the main block.

So:

- Main block: 6 cm × 6 cm × 5 cm = $6 \times 6 \times 5 = 180$ cm³
- Smaller block: 5 cm × 3 cm × 5 cm = $5 \times 3 \times 5 = 75$ cm³

✔ Total Volume = $180 + 75 = \boxed{255} \text{ cm}^3$

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✔ Final Answers:

| Solid | Volume |
|------|--------|
| 1 | 88 cm³ |
| 2 | 208 cm³ |
| 3 | 384 cm³ |
| 4 | 156 cm³ |
| 5 | 150 cm³ |
| 6 | 255 cm³ |

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📝 Summary of Solutions:

1. 88 cm³
2. 208 cm³
3. 384 cm³
4. 156 cm³
5. 150 cm³
6. 255 cm³

Let me know if you'd like a visual breakdown or explanation of any specific solid!