Volume of Composite Figures worksheet with four 3D shapes and calculation steps.

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 break each shape into simpler parts, calculate the volume of each part, and then add them together.

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a.

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

#### Part 1: Bottom prism
- Length = 9 m
- Width = 6 m
- Height = 9 m

$$
\text{Volume} = \text{length} \times \text{width} \times \text{height} = 9 \times 6 \times 9 = 486\ \text{m}^3
$$

#### Part 2: Top prism
- Length = 9 m
- Width = 6 m
- Height = 10 m (but it's only placed on top of a 9 m height, so the extra height is 10 - 9 = 1 m, but wait — actually, looking at the diagram: the top box is 10 m tall, and the bottom is 9 m tall, so the top box extends 1 m above the bottom one.

But the dimensions given:
- The top box has:
- Length = 9 m
- Width = 6 m
- Height = 10 m? Wait — no! Let’s read carefully.

Wait: the top box is labeled as:
- Height = 10 m? But the bottom box is 9 m high, and the top box sits on top.

Actually, the total height from bottom to top is 10 m, and the bottom box is 9 m, so the top box must be only 1 m high?

Wait — no! Look again.

The bottom box is labeled:
- Height = 9 m
- Length = 9 m
- Width = 6 m

The top box is sitting on top and is labeled:
- Height = 10 m? That can't be — that would make it taller than the whole structure.

Wait — the top box has:
- Height = 10 m? No — the label says "10 m" on the side of the top box, but the bottom box is 9 m high, so maybe the top box is 10 m long, not high.

Ah! Let's interpret the labels correctly.

Looking at the diagram:

- The bottom box:
- Length = 9 m (front-to-back)
- Width = 6 m (side-to-side)
- Height = 9 m (vertical)

- The top box:
- It's placed on top, with dimensions:
- Length = 9 m (same as bottom)
- Width = 6 m (same)
- Height = ? But the total vertical height of the entire shape is 10 m, and the bottom is 9 m, so the top box must be only 1 m high?

Wait — the label "10 m" is on the left side, going vertically — but the bottom box is 9 m, and the top box is 10 m? That doesn't make sense.

Wait — let's re-express:

Looking at the image:

- The bottom box: labeled with height 9 m, width 6 m, length 9 m
- The top box: placed on top, with height 10 m? But that would make it taller than the whole thing.

No — likely, the "10 m" is the length of the top box, not the height.

Wait — the diagram shows:

- On the left side, there is a vertical line labeled 10 m — that's the total height of the entire figure.
- Then, the bottom box is labeled 9 m in height, so the top box must be 10 - 9 = 1 m high?
- But the top box is shown as being shorter in width — no, actually, both have same width?

Wait — look at the side view: the top box is only 3 m wide, while the bottom is 6 m wide?

Wait — the top box has:
- Width = 3 m (labeled on the front)
- Length = 9 m (same as bottom)
- Height = ?

But the vertical dimension on the left side is labeled 10 m for the whole figure, and 9 m for the bottom box, so the top box must be 1 m high.

So:

#### Part 1: Bottom prism
- L = 9 m
- W = 6 m
- H = 9 m
- Volume = $9 \times 6 \times 9 = 486\ \text{m}^3$

#### Part 2: Top prism
- L = 9 m
- W = 3 m (since it's narrower)
- H = 10 - 9 = 1 m
- Volume = $9 \times 3 \times 1 = 27\ \text{m}^3$

#### Total Volume
$$
486 + 27 = \boxed{513}\ \text{m}^3
$$

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Now fill in the blanks:

> Volume of part 1:
> $9 \times 6 \times 9 = 486\ \text{m}^3$
>
> Volume of part 2:
> $9 \times 3 \times 1 = 27\ \text{m}^3$
>
> Volume of shape:
> $486 + 27 = \boxed{513}\ \text{m}^3$

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b.

This figure has three parts:
- A large base rectangle
- A smaller prism on top
- A small protrusion on the side?

Wait — looking at the diagram:

It looks like:
- A large horizontal base (like a shelf)
- A vertical prism on top of it
- And a smaller prism extending from the side?

But let's analyze.

From the diagram:
- The base is:
- Length = 8 m
- Width = 1 m
- Height = 3 m (the lower part)

Then on top of it:
- A vertical prism:
- Width = 5 m
- Depth = 6 m
- Height = 7 m

And also, there’s a smaller extension on the right side?

Wait — actually, looking at the labels:

- The horizontal base is:
- 8 m long
- 1 m high
- 3 m deep (into page)

- On top of it, there's a vertical block:
- 5 m wide
- 6 m deep
- 7 m high

But wait — the vertical block is on top of the base, and its base is 5 m x 6 m, which matches the top of the base?

Wait — the base is 8 m long, but the vertical block is only 5 m wide and 6 m deep, so it fits.

But is there another piece?

Wait — the horizontal base is 1 m high, and the vertical block is 7 m high, so it sits on top.

But is there a third piece?

No — actually, the entire shape is composed of:
1. A rectangular prism at the bottom (the base): 8 m × 1 m × 3 m
2. A tall rectangular prism on top: 5 m × 6 m × 7 m

Wait — but the depth of the base is 3 m, and the vertical block has depth 6 m — so it sticks out?

Wait — no, the depth is into the page. The base is 3 m deep, and the vertical block is 6 m deep — so it extends beyond?

But the vertical block is placed on the center of the base?

Wait — the base is 8 m long, 1 m high, 3 m deep.

The vertical block is:
- 5 m wide (in the length direction)
- 6 m deep (into page) — but base is only 3 m deep → so it extends 3 m beyond the base?

But the diagram shows the vertical block sitting on the base, so perhaps the base is 6 m deep, not 3 m?

Wait — the label says "3 cm" — but this is problem b, and units are meters? Wait — no, the labels say:

- For b: dimensions in meters? Wait — the labels are:
- 8 m, 6 m, 5 m, 7 m, 1 m, 3 m

But "3 cm" is written — no, it's "3 m" — typo in my reading.

Wait — the diagram says:

- Base: 8 m long, 1 m high, 3 m deep
- Vertical block: 5 m wide, 6 m deep, 7 m high

But the vertical block has 6 m depth, while the base has 3 m depth — so it sticks out behind?

But the diagram shows it sitting on top — so perhaps the base is 6 m deep?

Wait — the label on the right side says "3 m", which is the depth of the base.

But the vertical block is 6 m deep — so it must extend beyond.

But that's fine — it's a composite solid.

But wait — is the vertical block sitting on the entire base? Or just part?

Actually, the base is 8 m long, 3 m deep, 1 m high

The vertical block is:
- 5 m long (along the 8 m side)
- 6 m deep (so it extends 3 m beyond the base)
- 7 m high

So yes, it's placed on top of the base.

But is the vertical block attached to the base? Yes.

So the total volume is:

#### Part 1: Base
- L = 8 m
- W = 3 m (depth)
- H = 1 m
- Volume = $8 \times 3 \times 1 = 24\ \text{m}^3$

#### Part 2: Vertical block
- L = 5 m
- W = 6 m (depth)
- H = 7 m
- Volume = $5 \times 6 \times 7 = 210\ \text{m}^3$

But wait — is the vertical block sitting on the base? Yes — but does it overlap?

Yes — the base is 3 m deep, and the vertical block is 6 m deep, so the overlap is 3 m.

But since they are separate prisms, we just add volumes.

So total volume = $24 + 210 = \boxed{234}\ \text{m}^3$

But wait — is there a third part?

Wait — the vertical block is 7 m high, and the base is 1 m high, so it's sitting on top — correct.

But the vertical block is not resting entirely on the base? Its footprint is 5 m × 6 m, but the base is only 8 m × 3 m.

So the vertical block extends 3 m beyond the base in depth.

But that's okay — it's still valid.

So:

> Volume of part 1: $8 \times 3 \times 1 = 24\ \text{m}^3$
> Volume of part 2: $5 \times 6 \times 7 = 210\ \text{m}^3$
> Total volume: $24 + 210 = \boxed{234}\ \text{m}^3$

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c.

This is a T-shaped prism.

It has:
- A vertical column on the left
- A horizontal base extending to the right

Dimensions:
- The vertical part:
- Height = 11 cm
- Width = 4 cm
- Depth = 3 cm

- The horizontal part:
- Length = 8 cm
- Width = 3 cm
- Depth = 3 cm

But wait — the horizontal part is attached to the bottom of the vertical part.

But the vertical part is 11 cm high, and the horizontal part is 3 cm high? Wait — no.

Wait — the horizontal part is 3 cm high, and the vertical part is 11 cm high, so the vertical part sits on top of the horizontal part?

Wait — the diagram shows:

- A horizontal base: 8 cm long, 3 cm wide, 3 cm high
- On top of it, a vertical prism: 4 cm wide, 3 cm deep, 11 cm high

But the vertical prism is 4 cm wide, and the horizontal base is 8 cm long, so it's centered?

But the vertical prism is 11 cm high, and the horizontal base is 3 cm high, so it's on top.

But is the vertical prism resting on the horizontal base?

Yes.

So:

#### Part 1: Horizontal base
- L = 8 cm
- W = 3 cm
- H = 3 cm
- Volume = $8 \times 3 \times 3 = 72\ \text{cm}^3$

#### Part 2: Vertical prism
- L = 4 cm
- W = 3 cm
- H = 11 cm
- Volume = $4 \times 3 \times 11 = 132\ \text{cm}^3$

Total volume = $72 + 132 = \boxed{204}\ \text{cm}^3$

But wait — is the vertical prism overlapping with the base? Yes — but since it's on top, we add.

But note: the vertical prism is only 4 cm wide, and the base is 8 cm long, so it's centered.

So:

> Volume of part 1: $8 \times 3 \times 3 = 72\ \text{cm}^3$
> Volume of part 2: $4 \times 3 \times 11 = 132\ \text{cm}^3$
> Total: $72 + 132 = \boxed{204}\ \text{cm}^3$

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d.

This is a rectangular prism with a hole cut out.

Looks like:
- A large box
- With a rectangular hole removed from the center

Dimensions:
- Outer box:
- Length = 12 m
- Width = 10 m
- Height = 5 m

- Hole (cut-out):
- Length = 6 m
- Width = 2 m
- Height = 5 m (same height)

Wait — the hole goes through the full height?

Yes — it's a rectangular prism removed from the interior.

So:

#### Volume of outer box:
$$
12 \times 10 \times 5 = 600\ \text{m}^3
$$

#### Volume of hole:
$$
6 \times 2 \times 5 = 60\ \text{m}^3
$$

#### Total volume:
$$
600 - 60 = \boxed{540}\ \text{m}^3
$$

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Now summarize:

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Final Answers:

#### a.
- Volume of part 1: $9 \times 6 \times 9 = 486\ \text{m}^3$
- Volume of part 2: $9 \times 3 \times 1 = 27\ \text{m}^3$
- Volume of shape: $486 + 27 = \boxed{513}\ \text{m}^3$

#### b.
- Volume of part 1: $8 \times 3 \times 1 = 24\ \text{m}^3$
- Volume of part 2: $5 \times 6 \times 7 = 210\ \text{m}^3$
- Volume of shape: $24 + 210 = \boxed{234}\ \text{m}^3$

#### c.
- Volume of part 1: $8 \times 3 \times 3 = 72\ \text{cm}^3$
- Volume of part 2: $4 \times 3 \times 11 = 132\ \text{cm}^3$
- Volume of shape: $72 + 132 = \boxed{204}\ \text{cm}^3$

#### d.
- Volume of outer box: $12 \times 10 \times 5 = 600\ \text{m}^3$
- Volume of hole: $6 \times 2 \times 5 = 60\ \text{m}^3$
- Volume of shape: $600 - 60 = \boxed{540}\ \text{m}^3$

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✔ All solved!