Volume and Surface Area of Rectangular Prisms Worksheet | 7th ... - Free Printable

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 bottom part extends 4 cm out.

But the left block is 3 cm long, 3 cm wide, 2 cm high.

Then the extension is 4 cm long, 3 cm wide, but only 1 cm high? That doesn’t make sense.

Wait — looking at the diagram:

It appears:
- The main body is 7 cm long (3 + 4), 3 cm wide, 2 cm high?
- But the top has a small protrusion?

No — let’s look again.

Actually, this shape has:
- A long horizontal block of length 7 cm, width 3 cm, height 1 cm
- And a vertical block on the left of 3 cm length, 3 cm width, 1 cm height — but wait, the total height is 2 cm.

Wait — perhaps better to split vertically.

Actually:
- The bottom layer is 7 cm (length) × 3 cm (width) × 1 cm (height)
$$
V_1 = 7 \times 3 \times 1 = 21\ \text{cm}^3
$$

- The top layer is only 3 cm long (on the left), 3 cm wide, 1 cm high
$$
V_2 = 3 \times 3 \times 1 = 9\ \text{cm}^3
$$

- Total = $21 + 9 = \boxed{30}\ \text{cm}^3$

Yes, because the total height is 2 cm, and the left side is taller.

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5)

This is a "T" shaped or L-shaped structure.

Break into two parts:

- Vertical block: 5 cm (length) × 2 cm (width) × 2 cm (height)
$$
V_1 = 5 \times 2 \times 2 = 20\ \text{cm}^3
$$

- Horizontal block: 7 cm (length) × 3 cm (width) × 2 cm (height)? Wait — no.

Wait — dimensions:
- The horizontal arm is 7 cm long, 3 cm wide, 2 cm high?
- But the vertical arm is 5 cm long, 2 cm wide, 2 cm high.

But they overlap.

Better to think:

- The base is 7 cm (length) × 3 cm (width) × 2 cm (height) — but that would be too big.

Wait — looking at the diagram:

The shape has:
- A vertical block of 5 cm (depth?) × 2 cm (width) × 2 cm (height)
- A horizontal block extending 7 cm in length, 3 cm width, 2 cm height — but only the front is extended.

Actually, the horizontal part is 7 cm long, 3 cm wide, 2 cm high — but the vertical part is attached to it.

Wait — perhaps the vertical part is 5 cm long, 2 cm wide, 2 cm high, and the horizontal part is 7 cm long, 3 cm wide, 2 cm high — but they share a 2 cm × 2 cm face.

But we must avoid double-counting.

Let’s see:

- The horizontal block is 7 cm (length) × 3 cm (width) × 2 cm (height) → but the vertical block sits on top of it?

No — the diagram shows the vertical block is attached to the side.

Actually, it’s better to split into two cuboids:

1. Long horizontal base: 7 cm (length) × 3 cm (width) × 2 cm (height)
But wait — the vertical block is only 2 cm wide, so maybe:

Let’s define:

- Block A (horizontal): 7 cm (length) × 3 cm (width) × 2 cm (height)
$$
V_A = 7 \times 3 \times 2 = 42\ \text{cm}^3
$$

- Block B (vertical): This is a 5 cm long (in depth?), but it's only 2 cm wide and 2 cm high, and it's placed on top of the horizontal block.

Wait — the vertical block is 5 cm deep, 2 cm wide, 2 cm high, and it's attached to the end of the horizontal block.

But the horizontal block is 7 cm long, and the vertical block is 5 cm long — but they overlap?

Wait — no.

Looking at the diagram:

- The horizontal block is 7 cm long, 3 cm wide, 2 cm high.
- The vertical block is 5 cm long, 2 cm wide, 2 cm high — but it's placed perpendicular to the horizontal block, and its base overlaps with the end of the horizontal block.

But the total width is 3 cm, and the vertical block is only 2 cm wide — so it fits.

So:
- Horizontal block: 7 cm × 3 cm × 2 cm = 42 cm³
- Vertical block: 5 cm × 2 cm × 2 cm = 20 cm³

But do they overlap? Yes — the overlap region is 5 cm × 2 cm × 2 cm — but that's the entire vertical block.

Wait — no, the vertical block is not overlapping in volume — it’s attached to the side.

Actually, the vertical block is sitting on top of the end of the horizontal block.

But the horizontal block already has 2 cm height — and the vertical block is also 2 cm high, so it's just attached.

So no overlap in volume.

So total volume = $42 + 20 = \boxed{62}\ \text{cm}^3$

Wait — but the vertical block is only 5 cm long, and the horizontal block is 7 cm long — so if the vertical block is attached to the end, it's fine.

But is the vertical block on top of the horizontal block?

Yes — the diagram shows the vertical block is on top of the horizontal block.

But both have height 2 cm — so they are at the same level.

Wait — actually, the vertical block is on top of the horizontal block — but the horizontal block is 2 cm high, and the vertical block is also 2 cm high — so it's flush.

But the vertical block is only 2 cm wide, while the horizontal block is 3 cm wide — so it’s placed on the edge.

So total volume = $42 + 20 = \boxed{62}\ \text{cm}^3$

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6)

This is a large cuboid with a hole cut out.

- Outer dimensions: 8 cm (length) × 6 cm (width) × 4 cm (height)
$$
V_{\text{outer}} = 8 \times 6 \times 4 = 192\ \text{cm}^3
$$

- Hole: It's a rectangular hole, 4 cm (length) × 2 cm (width) × 4 cm (height)? Wait — the hole goes all the way through?

Looking at the diagram: The hole is inside, and it's shown as a rectangle of 4 cm by 2 cm, and it’s deep — but how deep?

The hole is cut into the solid, and its depth is equal to the height of the box? No — it’s shown as a rectangular prism inside.

Actually, the hole is 4 cm long, 2 cm wide, and 4 cm high? But the box is 4 cm high — so yes, it goes all the way through.

Wait — but the hole is not going through the full length — it’s only 4 cm long.

So the hole is a rectangular prism of:
- Length: 4 cm
- Width: 2 cm
- Height: 4 cm

So volume of hole = $4 \times 2 \times 4 = 32\ \text{cm}^3$

But wait — is the hole through the entire height? Yes — since it's shown as open on the top.

So:
- Volume of solid = outer volume - hole volume
$$
= 192 - 32 = \boxed{160}\ \text{cm}^3
$$

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✔ Final Answers:

1) $ \boxed{24} $ cm³
2) $ \boxed{84} $ cm³
3) $ \boxed{48} $ cm³
4) $ \boxed{30} $ cm³
5) $ \boxed{62} $ cm³
6) $ \boxed{160} $ cm³

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Let me know if you'd like explanations for surface area as well!