Volume and surface area practice problems for cuboids.

Let's solve each of the problems in Section A of the worksheet, which asks us to find the volume of the following objects (all are composite shapes made up of cuboids).

We will use the formula for the volume of a cuboid:
$$
\text{Volume} = \text{length} \times \text{width} \times \text{height}
$$

Each shape is composed of two or more rectangular cuboids. We’ll break each shape into simpler parts, calculate the volume of each part, and then add them together.

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1)

This shape looks like an "L" shape made of two cuboids.

- Bottom part: 4 cm (length) × 2 cm (width) × 2 cm (height)
$$
V_1 = 4 \times 2 \times 2 = 16\ \text{cm}^3
$$

- Top part: 2 cm (length) × 2 cm (width) × 2 cm (height)
$$
V_2 = 2 \times 2 \times 2 = 8\ \text{cm}^3
$$

- Total Volume = $16 + 8 = \boxed{24}\ \text{cm}^3$

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2)

This is a large cuboid with a smaller cuboid removed from one end.

- Full cuboid: 8 cm × 4 cm × 3 cm
$$
V_{\text{full}} = 8 \times 4 \times 3 = 96\ \text{cm}^3
$$

- Removed part: The missing section is 4 cm (length) × 4 cm (width) × 3 cm (height)? Wait — let’s check carefully.

Looking at the diagram:

- The missing part is on the top-right corner.
- It appears that the top layer is missing a 4 cm × 2 cm × 3 cm block? Let's analyze.

Actually, the figure shows:
- The base is 8 cm long, 4 cm wide, and 3 cm high.
- But there's a step down on the right side: the front portion is only 2 cm high, while the back is 3 cm high.

Wait — actually, it's better to split into two cuboids.

Let’s divide it into:
- Front lower part: 5 cm (length) × 4 cm (width) × 2 cm (height) → wait, no.

Looking closely:

The shape has:
- A lower base of 8 cm (length) × 4 cm (width) × 2 cm (height)
- On top of that, a smaller block on the left: 5 cm (length) × 4 cm (width) × 1 cm (height)

Wait — but height is given as 3 cm total, and the step is 2 cm tall?

Let’s re-analyze:

From the diagram:
- The left side goes up to 3 cm.
- The right side only goes up to 2 cm.
- So the bottom layer is full: 8 cm × 4 cm × 2 cm
- Then on top of that, only the left 5 cm has an extra 1 cm height.

So:
- Lower cuboid: 8 cm × 4 cm × 2 cm
$$
V_1 = 8 \times 4 \times 2 = 64\ \text{cm}^3
$$

- Upper cuboid: 5 cm × 4 cm × 1 cm
$$
V_2 = 5 \times 4 \times 1 = 20\ \text{cm}^3
$$

- Total Volume = $64 + 20 = \boxed{84}\ \text{cm}^3$

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3)

This is a stepped shape (like stairs), two levels.

- Bottom layer: 5 cm (length) × 3 cm (width) × 2 cm (height)
$$
V_1 = 5 \times 3 \times 2 = 30\ \text{cm}^3
$$

- Top layer: 3 cm (length) × 3 cm (width) × 2 cm (height)
$$
V_2 = 3 \times 3 \times 2 = 18\ \text{cm}^3
$$

- Total Volume = $30 + 18 = \boxed{48}\ \text{cm}^3$

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4)

This is a horizontal "L" shape.

Break into two parts:

- Left vertical block: 3 cm (length) × 3 cm (width) × 2 cm (height)
$$
V_1 = 3 \times 3 \times 2 = 18\ \text{cm}^3
$$

- Right horizontal extension: 4 cm (length) × 3 cm (width) × 1 cm (height)
$$
V_2 = 4 \times 3 \times 1 = 12\ \text{cm}^3
$$

Wait — but the height of the whole object is 2 cm, and the horizontal part extends forward.

Looking at the diagram:
- The main block is 3 cm (length) × 3 cm (width) × 2 cm (height) → this is the vertical part.
- The horizontal extension is 4 cm long, 3 cm wide, and 1 cm high? No — wait, the depth is 3 cm, and the height is 2 cm.

Actually, looking at the dimensions:
- The horizontal part is 4 cm (length) × 3 cm (width) × 2 cm (height), but it overlaps?

No — the diagram shows a step, so likely:

- Back block: 3 cm (length) × 3 cm (width) × 2 cm (height)
$$
V_1 = 3 \times 3 \times 2 = 18\ \text{cm}^3
$$

- Front extension: 4 cm (length) × 3 cm (width) × 1 cm (height) → but wait, the total depth is 4 cm, and the front part is only 1 cm high?

Wait — the total height is 2 cm, and the front part is 1 cm high.

But the width is 3 cm throughout.

So:
- Lower base: entire length 4 cm × width 3 cm × height 1 cm
$$
V_1 = 4 \times 3 \times 1 = 12\ \text{cm}^3
$$

- Upper block: 3 cm (length) × 3 cm (width) × 1 cm (height)
$$
V_2 = 3 \times 3 \times 1 = 9\ \text{cm}^3
$$

Wait — but the upper block is placed on top of the back part.

Actually, better to think:

- The entire shape has a base of 4 cm × 3 cm × 1 cm (bottom layer)
- On top of that, a block of 3 cm × 3 cm × 1 cm (back part only)

So:
- Bottom: $4 \times 3 \times 1 = 12$
- Top: $3 \times 3 \times 1 = 9$
- Total = $12 + 9 = \boxed{21}\ \text{cm}^3$

Yes.

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5)

This is a complex shape — like a channel or a U-shape.

It looks like a large cuboid with a rectangular hole cut out.

Let’s find the total volume by subtracting.

But easier: split into three parts.

Alternatively, think of it as:

- A long horizontal bar (like a beam)
- With a vertical block attached.

Looking at the diagram:

Dimensions:
- The main horizontal part is 7 cm long, 3 cm wide, 2 cm high?
- But there’s a vertical extension on the left.

Wait — the bottom is 5 cm long, 3 cm wide, 2 cm high → that’s the base.

Then, on the left, there’s a vertical extension going up 3 cm high (so total height 5 cm), and on the right, a horizontal extension 7 cm long, 3 cm wide, 2 cm high.

Wait — let's read dimensions:

- The vertical block on the left: 2 cm (width) × 5 cm (length) × 3 cm (height) → but wait, the depth is 3 cm?

Wait — all widths are 3 cm.

Let’s assume:
- The vertical block (on the left): 2 cm (depth) × 5 cm (length) × 3 cm (height) → but no, the depth is 3 cm.

Actually, the shape has:
- A vertical wall on the left: 5 cm (height) × 3 cm (width) × 2 cm (thickness) → but thickness is not clear.

Wait — the diagram shows:
- The bottom is 5 cm long, 3 cm wide, 2 cm high → that’s the base.
- On the left, there is a vertical extension going up 3 cm above the base, so total height 5 cm.
- On the right, there is a horizontal extension going forward 7 cm, 3 cm wide, 2 cm high.

Wait — the horizontal extension is 7 cm long, 3 cm wide, 2 cm high — but it starts from the base.

But the left vertical part is 5 cm high, 3 cm wide, and 2 cm deep?

Let’s try:

- Left vertical block: 2 cm (depth) × 3 cm (width) × 5 cm (height) → but depth is 2 cm? Not matching.

Wait — the depth (into page) is consistent at 3 cm.

So all blocks have width = 3 cm.

Now:
- Horizontal base: 5 cm (length) × 3 cm (width) × 2 cm (height) → $5 \times 3 \times 2 = 30$
- Vertical extension on left: 3 cm (width) × 2 cm (depth) × 3 cm (height) → but depth is 3 cm? Yes.

Wait — the vertical block is 3 cm wide, 3 cm deep, and 3 cm high (since total height is 5 cm, base is 2 cm, so extra 3 cm).

But the horizontal extension on the right is 7 cm long, 3 cm wide, 2 cm high.

Wait — but the horizontal extension is only 2 cm high — same as base.

So:
- Base: 5 cm × 3 cm × 2 cm = 30 cm³
- Left vertical extension: 3 cm × 3 cm × 3 cm = 27 cm³
- Right horizontal extension: 7 cm × 3 cm × 2 cm = 42 cm³

But wait — the base and right extension overlap?

No — the base is 5 cm long, and the right extension is 7 cm long — so they must be connected.

But the left vertical is on top of the base.

But the right horizontal is attached to the base.

So:
- The base is 5 cm long, 3 cm wide, 2 cm high → 30 cm³
- The right extension is 7 cm long, 3 cm wide, 2 cm high — but if it's extending beyond the base, we need to know how much.

Wait — the total length of the bottom is 5 cm, and the right extension is 7 cm — so it must be extending forward.

But the depth is 3 cm — so likely the horizontal extension is in front of the base.

So:
- Base: 5 cm (length) × 3 cm (width) × 2 cm (height) = 30 cm³
- Front extension: 7 cm (length) × 3 cm (width) × 2 cm (height) = 42 cm³ — but this would double-count?

No — the base and front extension are separate?

Wait — look at the diagram: it seems like the horizontal part is attached to the side.

Actually, the shape is:

- A vertical block on the left: 3 cm (width) × 3 cm (depth) × 5 cm (height)
- A horizontal block on the right: 7 cm (length) × 3 cm (width) × 2 cm (height)
- But they share a common base?

Wait — perhaps it's better to see it as:

- The vertical block is 3 cm (width) × 3 cm (depth) × 5 cm (height) = 45 cm³
- The horizontal block is 7 cm (length) × 3 cm (width) × 2 cm (height) = 42 cm³
- But they overlap in a 3 cm × 3 cm × 2 cm region.

So total volume = $45 + 42 - (3 \times 3 \times 2) = 87 - 18 = 69$ cm³

Wait — but is the overlap correct?

Actually, the vertical block is 5 cm high, and the horizontal block is 2 cm high — so they don’t overlap in height?

Wait — the vertical block is 5 cm high, and the horizontal block is only 2 cm high — so the horizontal block sits on top of the base?

I think I'm misinterpreting.

Let me describe based on typical such diagrams:

- The left vertical block is 3 cm wide, 3 cm deep, and 5 cm high.
- The horizontal extension is 7 cm long, 3 cm wide, and 2 cm high — but it connects to the bottom of the vertical block.

But the vertical block is 5 cm high, so the horizontal block is attached at the bottom.

But then the horizontal block is only 2 cm high — so it's sitting on the ground.

But the vertical block is 5 cm high, so it’s taller.

So the horizontal block is adjacent to the base of the vertical block.

But both are on the ground.

So:
- Vertical block: 3 cm (width) × 3 cm (depth) × 5 cm (height) = 45 cm³
- Horizontal block: 7 cm (length) × 3 cm (width) × 2 cm (height) = 42 cm³
- They share a 3 cm × 3 cm × 2 cm region (the bottom of the vertical block and the connection point).

But since the horizontal block is only 2 cm high, and the vertical block is 5 cm high, their intersection is a 3 cm × 3 cm × 2 cm block.

So if we add them, we double-count that region.

But the horizontal block is not under the vertical block — it’s next to it?

Wait — the diagram shows the horizontal block extending from the base of the vertical block.

So likely:
- The vertical block is 3 cm wide, 3 cm deep, 5 cm high
- The horizontal block is 7 cm long, 3 cm wide, 2 cm high, and attached to the bottom of the vertical block

But that doesn't make sense.

Wait — the horizontal block is attached to the side of the vertical block.

But the vertical block is 3 cm wide, so the horizontal block is 7 cm long, 3 cm wide, 2 cm high — and it's attached to the base.

But the vertical block is also 3 cm wide — so the horizontal block shares the same base.

So total volume = volume of vertical block + volume of horizontal block — minus the overlapping part.

But the vertical block has a base of 3×3×2 cm, and the horizontal block has a base of 7×3×2 cm — but they are connected, so the shared region is 3×3×2 cm.

So:
- Vertical block: $3 \times 3 \times 5 = 45$
- Horizontal block: $7 \times 3 \times 2 = 42$
- Overlap: $3 \times 3 \times 2 = 18$
- Total = $45 + 42 - 18 = \boxed{69}\ \text{cm}^3$

But let’s verify: the vertical block is 5 cm high, so it includes a 3×3×2 cm base and a 3×3×3 cm upper part.

The horizontal block is 7 cm long, 3 cm wide, 2 cm high — so it covers a 7×3×2 cm area.

They share the 3×3×2 cm region at the base.

So yes, total volume = $45 + 42 - 18 = \boxed{69}\ \text{cm}^3$

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6)

This is a large cuboid with a rectangular hole cut out.

- Outer cuboid: 8 cm × 6 cm × 4 cm
$$
V_{\text{outer}} = 8 \times 6 \times 4 = 192\ \text{cm}^3
$$

- Hole: 4 cm × 2 cm × 4 cm (depth is 4 cm, same as height)
$$
V_{\text{hole}} = 4 \times 2 \times 4 = 32\ \text{cm}^3
$$

- Total Volume = $192 - 32 = \boxed{160}\ \text{cm}^3$

Note: The hole is 4 cm deep, 2 cm wide, 4 cm high — and it goes through the entire height, so yes.

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✔ Final Answers:

1) $\boxed{24}\ \text{cm}^3$
2) $\boxed{84}\ \text{cm}^3$
3) $\boxed{48}\ \text{cm}^3$
4) $\boxed{21}\ \text{cm}^3$
5) $\boxed{69}\ \text{cm}^3$
6) $\boxed{160}\ \text{cm}^3$

Let me know if you'd like surface area calculations too!