Practice calculating the volume of composite shapes with this 'Volume of L-Blocks' worksheet, featuring nine distinct problems with varying dimensions.

Let's solve each of these L-shaped blocks by calculating their volume. Since all the shapes are made up of rectangular prisms (cuboids), we can break each L-block into two separate rectangular prisms, calculate their volumes individually, and then add them together.

The formula for the volume of a rectangular prism is:

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

We'll go through each one step by step.

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1)

- Dimensions:
- Left vertical part: 10 mm high, 8 mm wide, 2 mm deep
- Right horizontal part: 4 mm high (since total height is 10 mm, but bottom is 3 mm high), 12 mm long, 2 mm deep
Wait — actually, let’s re-analyze carefully.

Looking at the diagram:
- The entire base is 12 mm long.
- The left vertical section is 10 mm tall, 8 mm wide, and 2 mm deep.
- The right horizontal section extends from x=8 to x=12 → so 4 mm wide.
- Height of the right section is 3 mm (bottom layer).
- Depth is 2 mm throughout.

So split into:
- Vertical block: 10 mm × 8 mm × 2 mm
$$
V_1 = 10 \times 8 \times 2 = 160 \text{ mm}^3
$$
- Horizontal block: 3 mm × 4 mm × 2 mm
$$
V_2 = 3 \times 4 \times 2 = 24 \text{ mm}^3
$$
- Total Volume = 160 + 24 = 184 mm³

✔ Answer: 184 mm³

---

2)

Units: feet (ft)
- Vertical part: 16 ft high, 4 ft wide, 5 ft deep?
Wait — look closely:
- Bottom horizontal block: 5 ft (length), 4 ft (height), 8 ft (depth)? No — labeling is confusing.

From diagram:
- Bottom block: length = 5 ft, width = 8 ft, height = 4 ft
- Top block: attached vertically on the side — height = 16 ft - 4 ft = 12 ft? But it says 4 ft high? Wait...

Actually:
- The vertical part is 16 ft tall, 4 ft wide, and depth = ?
- The horizontal part is 8 ft long, 4 ft high, 5 ft deep?

Wait — better way: dimensions shown:
- Bottom block: 5 ft (length), 4 ft (height), 8 ft (depth)?
But label shows:
- Side: 5 ft
- Front: 8 ft
- Height: 4 ft
- Then top block: 4 ft (height), 4 ft (width), 16 ft (length)?

Wait — this is ambiguous.

Re-expressing:

From image:
- Horizontal base: 5 ft (length), 8 ft (depth), 4 ft (height)
- Vertical extension: 4 ft (width), 16 ft (height), 8 ft (depth)? But only the top part is shown.

Wait — the vertical block is 16 ft tall, but the base is only 4 ft high, so the vertical part is 12 ft tall?

But the label says "4 ft" on the vertical block — probably meaning width.

Let’s interpret:
- The bottom block: length = 5 ft, width = 8 ft, height = 4 ft
- The top block: attached to the end, same depth (8 ft), width = 4 ft, height = 16 ft – 4 ft = 12 ft?

No — the vertical block is labeled with 4 ft on its side and 16 ft on its front face.

Wait — likely:
- The vertical part has:
- Height = 16 ft
- Width = 4 ft
- Depth = 8 ft (same as bottom)

But bottom block is 5 ft long, 8 ft deep, 4 ft high.

But they share the 4 ft height? So:
- Bottom block: 5 ft × 8 ft × 4 ft = 160 ft³
- Vertical block: 4 ft × 8 ft × (16 - 4) = 4 × 8 × 12 = 384 ft³

But wait — does the vertical block extend full 16 ft? But it's only 4 ft wide.

Actually, the total height is 16 ft, and the bottom is 4 ft high, so the vertical part is 12 ft high.

But the vertical block is 4 ft wide, 12 ft high, and 8 ft deep.

So:
- Bottom: 5 ft × 8 ft × 4 ft = 160 ft³
- Vertical: 4 ft × 8 ft × 12 ft = 384 ft³

Total: 160 + 384 = 544 ft³

Wait — but the vertical block's width is 4 ft, and the bottom block is 5 ft long — do they align?

Yes — the vertical block is attached to the side of the bottom block.

So yes.

✔ Answer: 544 ft³

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3)

Units: cm
Shape: L-shape lying flat
Dimensions:
- Base: 9 cm long, 5 cm wide, 3 cm high
- Top extension: 1 cm high, 2 cm wide, 9 cm long? But that would be over the whole thing.

Wait — looking:
- The top part is 1 cm high, 2 cm wide, and spans the length? But it's only on one end.

Actually:
- The main block: 9 cm × 5 cm × 3 cm → but the top is only 1 cm high on one end.

Wait — the shape has:
- A lower rectangle: 9 cm (length), 5 cm (width), 3 cm (height)
- On top of that, a smaller rectangle: 1 cm (height), 2 cm (width), 9 cm (length)? No — but the top is only 2 cm wide.

Wait — the top piece is 1 cm high, 2 cm wide, and goes along the full 9 cm? But the diagram shows it's only on one side.

Wait — no: the entire base is 9 cm, and the top is 2 cm wide, but the side is 1 cm high.

Wait — better:
- The bottom block: 9 cm (length), 5 cm (width), 3 cm (height)
- The top block: 2 cm (width), 1 cm (height), 9 cm (length)? But that would make it wider than the base.

Wait — no. Looking at the figure:
- The top block is 2 cm wide (in the direction of the 5 cm side), 1 cm high, and 9 cm long (along the 9 cm direction).

But the base is 5 cm wide, so if the top is 2 cm wide, it must be extending outward.

But the diagram shows it's on top of the main block.

Wait — perhaps:
- The main body: 9 cm (length), 5 cm (width), 3 cm (height)
- The top extension: 2 cm (width), 1 cm (height), 9 cm (length)

But the top extension is only on one end? Or the whole length?

Actually, the top piece is only 2 cm wide and 1 cm high, and it sits on top of the front edge of the base.

But the base is 5 cm wide, so the top piece is 2 cm wide, so it doesn't cover the full width.

But the label says “1 cm” on top, “2 cm” on side, “9 cm” on bottom.

So:
- The top block: 9 cm × 2 cm × 1 cm = 18 cm³
- The bottom block: 9 cm × 5 cm × 3 cm = 135 cm³

But wait — the bottom block already includes the area under the top block?

No — the top block is on top, so we just add.

But the bottom block is 9×5×3 = 135 cm³

Top block: 9×2×1 = 18 cm³

Total: 135 + 18 = 153 cm³

But is the top block sitting on the 5 cm side? It's only 2 cm wide, so it's like a ridge.

Yes — so total volume is sum.

✔ Answer: 153 cm³

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4)

Units: inches
Shape: Two steps
Dimensions:
- Bottom block: 12 in × 10 in × 4 in
- Top block: 10 in × 3 in × 5 in

Wait — check:
- Bottom: length = 12 in, width = 10 in, height = 4 in → V = 12×10×4 = 480 in³
- Top: placed on top, length = 10 in, width = 3 in, height = 5 in → but wait, height is 5 in, but bottom is 4 in — so total height is 4+5=9 in?

But the diagram shows the top block is 5 in high, and bottom is 4 in high — so yes.

But the top block is only 10 in long, 3 in wide, 5 in high.

But is it sitting on the back or side?

From diagram:
- Bottom block: 12 in long, 10 in wide, 4 in high
- Top block: 10 in long, 3 in wide, 5 in high — but placed on the back of the bottom block?

But the label shows 3 in on the side — so width = 3 in.

But the top block is 10 in long, so it matches the width of the bottom.

So:
- Bottom: 12 × 10 × 4 = 480 in³
- Top: 10 × 3 × 5 = 150 in³

Total: 480 + 150 = 630 in³

✔ Answer: 630 in³

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5)

Units: meters
Shape: L-shaped
- Long horizontal arm: 15 m × 4 m × 11 m? Wait — no.

Look:
- The long arm: 15 m (length), 4 m (height), 11 m (depth)? But depth is not given.

Wait — labels:
- Length: 15 m
- Height: 4 m
- Depth: 2 m? Wait — the short arm has 2 m depth.

Wait — the short arm is 2 m wide, 5 m long, 11 m high?

Wait — diagram:
- One arm: 15 m long, 4 m high, 2 m deep
- Other arm: 5 m long, 11 m high, 2 m deep

But both have 2 m depth?

Wait — the depth is consistent.

So:
- First block: 15 m × 4 m × 2 m = 120 m³
- Second block: 5 m × 11 m × 2 m = 110 m³

Total: 120 + 110 = 230 m³

But are they connected?

Yes — the 15 m arm is horizontal, the 5 m arm is vertical, both 2 m deep.

So yes.

✔ Answer: 230 m³

---

6)

Units: cm
Shape: L-shaped, standing
- Vertical block: 16 cm high, 2 cm wide, 8 cm deep
- Horizontal block: 3 cm high, 8 cm long, 2 cm deep

But the vertical block is 16 cm high, 2 cm wide, 8 cm deep

Bottom block: 3 cm high, 8 cm long, 2 cm deep

But they overlap? No — the vertical block is on top of the horizontal one?

Wait — the diagram shows:
- Bottom block: 8 cm (length), 3 cm (height), 2 cm (depth)
- Top block: 16 cm (height), 2 cm (width), 8 cm (length)? But 16 cm high?

Wait — the vertical part is 16 cm tall, but the bottom is only 3 cm high, so the vertical part is 16 - 3 = 13 cm tall?

But the label says "16 cm" on the side — probably total height.

Wait — no — the vertical block is 16 cm tall, 2 cm wide, 8 cm deep

But the horizontal block is 3 cm high, 8 cm long, 2 cm deep

And they are joined at the corner.

But the horizontal block is 3 cm high, and the vertical block is 16 cm high — so the vertical block is taller.

But the horizontal block is only 3 cm high — so the vertical block is on top of it?

Wait — the vertical block is 2 cm wide, and the horizontal block is 2 cm deep — so they match.

So:
- Horizontal block: 8 cm × 3 cm × 2 cm = 48 cm³
- Vertical block: 2 cm × 16 cm × 8 cm? Wait — no.

Wait — the vertical block is 16 cm tall, 2 cm wide, and 8 cm deep? But the depth is 8 cm — same as length.

But the horizontal block is 8 cm long, 3 cm high, 2 cm deep.

So:
- Horizontal: 8 × 3 × 2 = 48 cm³
- Vertical: 16 × 2 × 8 = 256 cm³

But are they overlapping? Yes — the vertical block is on top of the horizontal block, but only over a portion.

Wait — the vertical block is 2 cm wide, 16 cm high, and 8 cm deep — but the horizontal block is only 2 cm deep.

Wait — the horizontal block is 8 cm long, 3 cm high, 2 cm deep

Vertical block: 2 cm wide, 16 cm high, 8 cm deep — but depth is 8 cm, while horizontal is only 2 cm deep.

Wait — confusion.

Looking at diagram:
- The horizontal block: length = 8 cm, height = 3 cm, depth = 2 cm → V = 8×3×2 = 48 cm³
- The vertical block: height = 16 cm, width = 2 cm, depth = 8 cm? But depth is 8 cm, but the horizontal block is only 2 cm deep.

Wait — no — the depth of the vertical block should be 2 cm to match.

Wait — the diagram shows:
- The vertical block has 2 cm on the side — probably depth.
- The horizontal block has 2 cm on the side — depth.

So both have 2 cm depth.

Then:
- Vertical block: height = 16 cm, width = 2 cm, depth = 2 cm → V = 16×2×2 = 64 cm³
- Horizontal block: length = 8 cm, height = 3 cm, depth = 2 cm → V = 8×3×2 = 48 cm³

But are they connected?

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

But the horizontal block is 8 cm long, and the vertical block is 16 cm tall — so the vertical block is standing on the end of the horizontal block.

So total volume = 64 + 48 = 112 cm³

✔ Answer: 112 cm³

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7)

Units: feet
Shape: L-shaped, horizontal
- Long arm: 11 ft long, 6 ft wide, 3 ft high
- Short arm: 5 ft high, 2 ft wide, 6 ft long? But wait.

Wait — diagram:
- Main block: 11 ft long, 6 ft wide, 3 ft high
- Attached on top: 5 ft high, 2 ft wide, 6 ft long? But 5 ft high — but main block is 3 ft high, so extra 2 ft?

Wait — the top block is 5 ft high, but the main block is 3 ft high — so the top block is taller.

But the top block is only 2 ft wide.

So:
- Bottom block: 11 ft × 6 ft × 3 ft = 198 ft³
- Top block: 6 ft × 2 ft × 5 ft = 60 ft³

But are they aligned?

Yes — the top block is on top of the main block, but only over a 6 ft length.

But the main block is 11 ft long — so the top block is on one end.

So total volume = 198 + 60 = 258 ft³

✔ Answer: 258 ft³

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8)

Units: inches
This one is tricky — looks like a U-shape or channel.

Dimensions:
- Outer shape: 18 in wide, 12 in deep, 5 in high?
- But there’s a cutout.

Wait — it’s an L-block but turned.

From diagram:
- It appears to be a large block with a notch.

But let’s see:
- The outer dimensions: 18 in (length), 12 in (depth), 5 in (height)
- But there’s a missing part: a rectangular hole of 6 in × 2 in × 12 in?

Wait — the cutout is 6 in wide, 2 in high, 12 in deep?

But the whole block is 18 in long, 12 in deep, 5 in high.

But the cutout is 6 in wide, 2 in high, 12 in deep — but depth is 12 in, which matches.

So:
- Full block: 18 × 12 × 5 = 1080 in³
- Cutout: 6 × 2 × 12 = 144 in³
- Volume = 1080 - 144 = 936 in³

Alternatively, think as two parts:
- Left block: 6 in × 12 in × 5 in = 360 in³
- Right block: 6 in × 12 in × 5 in = 360 in³
- Middle cutout: 6 in × 2 in × 12 in = 144 in³

Wait — no — the middle is missing.

Actually, the block is like a "U" shape — three parts?

Wait — no — it's an L-block rotated.

Better: treat as a big rectangle minus a smaller one.

But the diagram shows:
- The main block is 18 in long, 12 in deep, 5 in high
- A rectangular hole of 6 in wide, 2 in high, 12 in deep — but it's removed from the center?

Wait — the label shows:
- 18 in across
- 12 in depth
- 5 in height
- And a cutout of 6 in wide, 2 in high, 12 in deep

But the cutout is 6 in wide — so from the center?

But the shape is symmetric?

Wait — the top is 12 in deep, but the cutout is 12 in deep — so it's a trench.

But the height is 5 in, and the cutout is 2 in high — so it's a 2 in deep trench.

So:
- Volume of solid = (18 × 12 × 5) - (6 × 12 × 2) = 1080 - 144 = 936 in³

✔ Answer: 936 in³

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9)

Units: meters
Shape: L-shaped, stepped
Dimensions:
- Bottom block: 13 m long, 6 m wide, 7 m high
- Top block: 4 m high, 11 m long, 6 m wide? But width is 6 m, same.

Wait — the top block is 4 m high, but the bottom is 7 m high — so top is shorter?

But the top block is 11 m long, and bottom is 13 m long — so it's offset.

So:
- Bottom block: 13 m × 6 m × 7 m = 546 m³
- Top block: 11 m × 6 m × 4 m = 264 m³

Are they stacked?

Yes — the top block is on top of the bottom block, but only over a 11 m length.

So total volume = 546 + 264 = 810 m³

✔ Answer: 810 m³

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## ✔ Final Answers:

| # | Volume |
|---|--------|
| 1 | 184 mm³ |
| 2 | 544 ft³ |
| 3 | 153 cm³ |
| 4 | 630 in³ |
| 5 | 230 m³ |
| 6 | 112 cm³ |
| 7 | 258 ft³ |
| 8 | 936 in³ |
| 9 | 810 m³ |

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