Volume of Composite Figures Additive Volume - Worksheets - Free Printable

Let's solve each composite figure step by step. These are all composite solids made up of two or more rectangular prisms (boxes). To find the total volume, we:

1. Break the shape into simpler rectangular prisms.
2. Calculate the volume of each part using:
$$
\text{Volume} = \text{length} \times \text{width} \times \text{height}
$$
3. Add the volumes together.

---

a.

This shape consists of two rectangular prisms stacked on top of each other.

#### Part 1 (Bottom prism):
- Length = 9 m
- Width = 6 m
- Height = 9 m
$$
\text{Volume}_1 = 9 \times 6 \times 9 = 486 \text{ m}^3
$$

#### Part 2 (Top prism):
- Length = 9 m
- Width = 6 m
- Height = 10 m? Wait — no! The top is only 3 m high, and it sits on the bottom one.

But wait — looking closely:
- The top box has dimensions:
- Length = 9 m
- Width = 6 m
- Height = 3 m? No — actually, the height from the base to the top is 10 m, but the bottom box is 9 m high, so the top box must be 1 m high?

Wait — let’s re-express the image carefully.

Actually, based on the diagram:

- The bottom prism:
- Length = 9 m
- Width = 6 m
- Height = 9 m → Volume = $9 \times 6 \times 9 = 486$ m³

- The top prism:
- It extends 3 m above the bottom one, so its height is 3 m
- Its base is 6 m wide, and length is 6 m (since it's smaller and centered)
- But wait — the top box appears to have a length of 6 m, width of 6 m, and height of 3 m?
- However, the depth (width) is shown as 3 m in the side view?

Wait — let's interpret carefully.

From the diagram:

- The bottom block:
- Front face shows 9 m (height), 6 m (width), and 9 m (length)
- So: $ V_1 = 9 \times 6 \times 9 = 486 $ m³

- The top block:
- It is placed on top, with a height of 3 m (from the top of the bottom block to the very top)
- The width of the top block is 6 m (same as bottom)
- The depth (into the page) is 3 m? Wait — the side view shows the top block is 3 m deep, but the bottom block is 6 m deep?

Wait — the side view shows:
- Bottom block: 9 m tall, 6 m wide
- Top block: 3 m tall, 3 m wide (in the side view)

But the front view shows:
- Bottom block: 9 m long, 9 m tall
- Top block: 6 m long, 3 m tall

So likely:
- The bottom prism:
- Length = 9 m
- Width = 6 m
- Height = 9 m → $9 \times 6 \times 9 = 486$ m³

- The top prism:
- Length = 6 m
- Width = 3 m
- Height = 3 m → $6 \times 3 \times 3 = 54$ m³

Wait — this seems inconsistent.

Alternatively, perhaps the top block is 6 m long, 6 m wide, and 3 m high, but only partially over the bottom?

But the diagram shows the top block is 6 m long, 6 m wide, and 3 m high, sitting on top of the bottom block.

But the bottom block is 9 m long, 6 m wide, and 9 m high.

So the top block is centered on the bottom block.

So:

#### Volume of Part 1 (bottom):
- $9 \times 6 \times 9 = 486$ m³

#### Volume of Part 2 (top):
- $6 \times 6 \times 3 = 108$ m³

Total volume:
$$
486 + 108 = 594 \text{ m}^3
$$

But wait — the top block is only 6 m long, not 9 m, so it fits on top.

Yes.

So:

a.
- Volume of part 1: $9 \times 6 \times 9 = 486$ m³
- Volume of part 2: $6 \times 6 \times 3 = 108$ m³
- Total: $486 + 108 = \boxed{594}$ m³

---

b.

This is a L-shaped solid made of two rectangles.

We can split it into two parts.

#### Part 1: The vertical tower (left side)
- Height = 6 m
- Depth = 3 m
- Width = 5 m → so volume = $6 \times 3 \times 5 = 90$ m³

Wait — let’s read the labels:

From the diagram:

- The vertical block:
- Height = 6 m
- Width = 5 m
- Depth = 3 m → $6 \times 5 \times 3 = 90$ m³

- The horizontal base:
- Length = 8 m
- Width = 1 m
- Depth = 3 m → $8 \times 1 \times 3 = 24$ m³

Wait — but the horizontal base is attached to the bottom of the vertical block.

But the vertical block is 6 m high, and the horizontal base is 1 m high, so total height at the back is 7 m?

Wait — look at the diagram:

- The vertical block: 6 m high, 5 m wide, 3 m deep
- The horizontal block: 8 m long, 1 m high, 3 m deep — but it's attached to the bottom of the vertical block?

But then the total height would be 6 m at the front and 1 m at the back?

Wait — no. Actually, the horizontal block is 1 m high, and the vertical block is 6 m high, but they are connected at the same base level.

Looking at the diagram:

- The horizontal base is 1 m high, 8 m long, 3 m deep
- On top of that, there's a vertical block that is 5 m wide, 6 m high, and 3 m deep, but only extending 5 m along the length?

Wait — the vertical block is 5 m wide, 6 m high, and 3 m deep, and it's sitting on top of the horizontal base.

But the horizontal base is 8 m long, and the vertical block is only 5 m long? Or is it?

Actually, the vertical block is 5 m wide, 6 m high, and 3 m deep, and it's placed on the back of the horizontal base.

The horizontal base is 8 m long, 1 m high, 3 m deep

But the vertical block is 5 m wide (along the length), so it's only covering 5 m of the 8 m length.

So:

#### Part 1: Horizontal base
- $8 \times 1 \times 3 = 24$ m³

#### Part 2: Vertical tower
- $5 \times 6 \times 3 = 90$ m³

But wait — the vertical tower is 6 m high, and the base is only 1 m high, so the total height is 7 m at the back.

Yes.

So total volume:
$$
24 + 90 = \boxed{114} \text{ m}^3
$$

But wait — is the vertical tower sitting on top of the base? Yes.

So yes, total volume is sum.

But let’s double-check dimensions.

Labeling:

- Horizontal block: length = 8 m, width = 1 m, depth = 3 m → $8 \times 1 \times 3 = 24$
- Vertical block: length = 5 m, height = 6 m, depth = 3 m → $5 \times 6 \times 3 = 90$

Total: $24 + 90 = \boxed{114}$ m³

---

c.

This is a rectangular prism with a smaller prism cut out from the top.

Wait — no. It looks like a large rectangular prism with a smaller prism attached on top?

No — look: it's a tall prism with a smaller prism on top.

Wait — actually, it's a single rectangular prism with a smaller rectangular prism on top.

Dimensions:

- Large prism (bottom): height = 11 cm, width = 4 cm, depth = 3 cm → $11 \times 4 \times 3 = 132$ cm³

- Small prism (top): height = 3 cm, width = 4 cm, depth = 3 cm → $3 \times 4 \times 3 = 36$ cm³

Wait — but the small prism is on top, and it's 3 cm high, so total height is 11 + 3 = 14 cm?

But the label says "11 cm" for the large one, and "3 cm" for the small one.

But the width of both is 4 cm, depth is 3 cm.

So:

- Volume of part 1 (bottom): $11 \times 4 \times 3 = 132$ cm³
- Volume of part 2 (top): $3 \times 4 \times 3 = 36$ cm³
- Total: $132 + 36 = \boxed{168}$ cm³

Wait — but is the top prism attached? Yes.

But look at the diagram: the top prism is 3 cm high, and it's placed on top of the 11 cm one, so yes.

But is the top prism the same width and depth? Yes — 4 cm wide, 3 cm deep.

So total volume: $132 + 36 = \boxed{168}$ cm³

---

d.

This is a large rectangular prism with a smaller rectangular prism removed from the center.

So we need to:

1. Find volume of full outer prism
2. Subtract volume of the missing part

Outer dimensions:
- Length = 12 m
- Width = 10 m
- Height = 5 m → $12 \times 10 \times 5 = 600$ m³

Missing part (cut-out):
- Length = 6 m
- Width = 2 m
- Height = 5 m → $6 \times 2 \times 5 = 60$ m³

So volume of shape:
$$
600 - 60 = \boxed{540} \text{ m}^3
$$

---

✔ Final Answers:

#### a.
- Volume of part 1: $9 \times 6 \times 9 = 486$ m³
- Volume of part 2: $6 \times 6 \times 3 = 108$ m³
- Total: $486 + 108 = \boxed{594}$ m³

#### b.
- Volume of part 1 (base): $8 \times 1 \times 3 = 24$ m³
- Volume of part 2 (tower): $5 \times 6 \times 3 = 90$ m³
- Total: $24 + 90 = \boxed{114}$ m³

#### c.
- Volume of part 1: $11 \times 4 \times 3 = 132$ cm³
- Volume of part 2: $3 \times 4 \times 3 = 36$ cm³
- Total: $132 + 36 = \boxed{168}$ cm³

#### d.
- Volume of whole: $12 \times 10 \times 5 = 600$ m³
- Volume of hole: $6 \times 2 \times 5 = 60$ m³
- Total: $600 - 60 = \boxed{540}$ m³

---

✔ Summary:

| Problem | Volume |
|--------|--------|
| a | 594 m³ |
| b | 114 m³ |
| c | 168 cm³ |
| d | 540 m³ |

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