Student works on a volume calculation problem involving L-blocks on a math worksheet.

Let's solve the problem of finding the volume of each L-shaped block (also called an "L-block") step by step. The volume of a 3D shape like this is calculated by breaking it into simpler rectangular prisms, calculating their volumes individually, and then adding them together.

We'll go through each figure one at a time.

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Problem 1:

The first L-block has dimensions labeled:

- Height (h): 10 mm
- Length (l): 12 mm
- Width (w): 2 mm

But we also see that the vertical part is 8 mm in length (the horizontal arm) and 10 mm high, and the horizontal part is 12 mm long, 3 mm wide, and 2 mm tall.

Wait — let’s interpret this carefully.

Looking closely:
- The vertical arm (left side) is:
- Height = 10 mm
- Depth (into page) = 2 mm
- Width = 8 mm (since the horizontal part extends only 8 mm from the base)
- The horizontal arm (right side) is:
- Length = 12 mm
- Width = 3 mm (depth)
- Height = 2 mm (only the bottom layer)

But wait — actually, both arms have same depth (width into the page), which appears to be 2 mm, and the total width of the horizontal part is 3 mm? That doesn't match.

Let’s re-analyze based on standard interpretation.

Actually, from the diagram:
- The entire L-shape has:
- A vertical rectangle: height = 10 mm, width = 8 mm, depth = 2 mm
- A horizontal rectangle: length = 12 mm, width = 3 mm, depth = 2 mm
- But they overlap in the corner.

Wait — more likely, the total structure can be split into two rectangles:

Let’s define:

#### Part A: Vertical section
- Height = 10 mm
- Width = 8 mm (from the label)
- Depth = 2 mm (into the page)

So Volume A = 10 × 8 × 2 = 160 mm³

#### Part B: Horizontal extension
- Length = 12 mm
- Width = 3 mm (this is the depth, but wait — labels say 3 mm and 2 mm)

Wait — let's read the labels clearly:

From the image:
- The horizontal arm has:
- Length = 12 mm
- Width = 3 mm (side view)
- Height = 2 mm (but the vertical part is taller)

Wait — no. Actually, the height of the horizontal arm is 2 mm, while the vertical arm is 10 mm tall.

But the depth (into the page) for both parts seems to be 2 mm?

No — the label says "2 mm" on the bottom, which is likely the depth (into the page). So both arms have depth = 2 mm.

Now, the vertical arm:
- Height = 10 mm
- Width = 8 mm (from the side)
- Depth = 2 mm
→ Volume = 10 × 8 × 2 = 160 mm³

The horizontal arm:
- Length = 12 mm
- Width = 3 mm (but wait — the width should be consistent)

Wait — actually, the horizontal arm is 12 mm long, 3 mm wide, and 2 mm deep?

But then the overlap is where they meet.

Wait — perhaps better to think of it as:

The entire base is 12 mm long, and the height varies.

Alternatively, break it into two rectangles:

1. Vertical block: 8 mm (width) × 10 mm (height) × 2 mm (depth)
→ Volume = 8 × 10 × 2 = 160 mm³

2. Horizontal block: 12 mm (length) × 3 mm (width) × 2 mm (depth)? No — that would be too big.

Wait — look at the bottom:
- The horizontal arm is 12 mm long, 3 mm wide (in the front), and 2 mm deep.
- But the vertical arm is 8 mm wide, 10 mm tall, and 2 mm deep.

But they overlap in the corner.

So the correct way is to divide the L-shape into two rectangles:

- Rectangle 1: Top-left vertical part
- Dimensions: 8 mm (width) × 10 mm (height) × 2 mm (depth)
→ Volume = 8 × 10 × 2 = 160 mm³

- Rectangle 2: Bottom-right horizontal part
- Dimensions: (12 - 8) = 4 mm (width) × 2 mm (height) × 2 mm (depth)
→ Wait — the height of the horizontal part is 2 mm, not 10 mm.

But the vertical part goes up to 10 mm, and the horizontal part is only 2 mm high.

So actually, the horizontal arm is:
- Length = 12 mm
- Width = 3 mm? No — the depth is 2 mm.

Wait — confusion in labeling.

Let me clarify:

From the image:

- The vertical arm:
- Height = 10 mm
- Width = 8 mm (horizontal dimension)
- Depth = 2 mm (into the page)

- The horizontal arm:
- Length = 12 mm
- Height = 2 mm
- Depth = 2 mm
- But it overlaps with the vertical arm in the corner.

Wait — the total length of the base is 12 mm, and the vertical arm is 8 mm wide, so the horizontal arm extends 12 mm in length, but only (12 - 8) = 4 mm beyond the vertical arm.

But the horizontal arm has height 2 mm, and depth 2 mm, and length 12 mm.

But then the vertical arm is 8 mm wide, 10 mm high, 2 mm deep, and sits on top of the horizontal arm?

No — the vertical arm is attached to the end of the horizontal arm.

Better to visualize:

The L-shape consists of:

1. A tall vertical rectangle: 8 mm wide, 10 mm high, 2 mm deep
2. A short horizontal rectangle: 12 mm long, 2 mm high, 2 mm deep — but only the part not overlapping with the vertical one.

But if the vertical one is 8 mm wide, and the horizontal one is 12 mm long, then the overlapping region is 8 mm × 2 mm × 2 mm.

So total volume = Volume of vertical + Volume of horizontal – overlap

But since they are joined, we can compute:

- Vertical block: 8 mm × 10 mm × 2 mm = 160 mm³
- Horizontal block: (12 mm – 8 mm) = 4 mm wide, but wait — no.

Wait — actually, the horizontal arm is 12 mm long, but only extends beyond the vertical arm by 4 mm? Or is it 12 mm long, and the vertical arm is attached to one end?

Yes — the vertical arm is 8 mm wide, and the horizontal arm is 12 mm long, so the total base is 12 mm.

But the vertical arm is 8 mm wide, so the horizontal arm must extend 12 mm in length, but only (12 - 8) = 4 mm is extra?

No — the horizontal arm is 12 mm long, and the vertical arm is attached to its end, extending upward.

So the horizontal arm is: 12 mm (length) × 2 mm (height) × 2 mm (depth) = 48 mm³

The vertical arm is: 8 mm (width) × 10 mm (height) × 2 mm (depth) = 160 mm³

But do they overlap?

Yes — in the corner, there is a common region: 8 mm × 2 mm × 2 mm = 32 mm³

But since both blocks share that region, we must avoid double-counting.

So total volume = (Volume of vertical) + (Volume of horizontal) – (Overlap)

= 160 + 48 – 32 = 176 mm³

But wait — is the horizontal arm really 12 mm long? Let's check the diagram.

Looking at the image:
- The horizontal arm has length 12 mm
- The vertical arm is 8 mm wide
- The vertical arm is placed at the end of the horizontal arm, so the horizontal arm extends 12 mm, and the vertical arm is 8 mm wide, so the overlap is 8 mm × 2 mm × 2 mm

But the vertical arm has height 10 mm, and the horizontal arm has height 2 mm, so the overlap is only 2 mm high.

So yes, the overlapping region is: 8 mm × 2 mm × 2 mm = 32 mm³

So:

- Vertical block: 8 × 10 × 2 = 160 mm³
- Horizontal block: 12 × 2 × 2 = 48 mm³
- Overlap: 8 × 2 × 2 = 32 mm³

Total volume = 160 + 48 - 32 = 176 mm³

But wait — is the horizontal arm really 12 mm long and 2 mm high? Yes.

But another way: split the L-shape into two non-overlapping rectangles.

1. Bottom rectangle: 12 mm (length) × 2 mm (height) × 2 mm (depth) = 48 mm³
2. Top rectangle: 8 mm (width) × (10 - 2) = 8 mm × 8 mm × 2 mm = 128 mm³

Because the vertical arm is 10 mm tall, but 2 mm of that is already included in the bottom.

So total volume = 48 + 128 = 176 mm³

Same result.

✔ Answer for Problem 1: 176 mm³

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Problem 2:

Dimensions:
- The L-shape has:
- One arm: 8 ft × 5 ft × 4 ft
- Other arm: 16 ft × 4 ft × 4 ft
- They join at the corner.

Wait — let's read the labels:

- The horizontal arm: length = 8 ft, width = 5 ft, height = 4 ft
- The vertical arm: length = 16 ft, width = 4 ft, height = 4 ft

But they are connected — the vertical arm is attached to the end of the horizontal arm.

So:

- Horizontal block: 8 ft × 5 ft × 4 ft = 160 ft³
- Vertical block: 16 ft × 4 ft × 4 ft = 256 ft³

But they overlap in the corner: 4 ft × 4 ft × 4 ft = 64 ft³

So total volume = 160 + 256 - 64 = 352 ft³

Alternatively, split into two rectangles without overlap:

- Bottom part: 8 ft × 5 ft × 4 ft = 160 ft³
- Top part: (16 - 8) = 8 ft × 4 ft × 4 ft = 128 ft³? No — the vertical arm is 16 ft long, but only 8 ft extends beyond?

Wait — no. The vertical arm is 16 ft long, but the horizontal arm is 8 ft long, and they connect.

But the vertical arm is 4 ft wide, and the horizontal arm is 5 ft wide.

So the overlap is: 4 ft × 4 ft × 4 ft = 64 ft³

So total volume = 160 + 256 - 64 = 352 ft³

✔ Answer for Problem 2: 352 ft³

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Problem 3:

Dimensions:
- Base: 9 cm long, 2 cm high, 3 cm deep
- On top, a smaller block: 1 cm high, 5 cm long, 3 cm deep

Wait — let’s read:

- The main horizontal block: 9 cm long, 2 cm high, 3 cm deep → Volume = 9 × 2 × 3 = 54 cm³
- The top block: 1 cm high, 5 cm long, 3 cm deep → Volume = 5 × 1 × 3 = 15 cm³

Are they aligned? Yes — the top block is centered or offset?

From the diagram: the top block is on top of the main block, and extends over it.

But since both have depth = 3 cm, and the top block is 5 cm long, and the main block is 9 cm long, the overlap is 5 cm × 3 cm × 1 cm? No — the height of the overlap is 1 cm (the top block), and the base is 5 cm × 3 cm.

But the main block is 2 cm high, so the top block sits on top of it, and does not overlap in height.

So no double-counting — they are stacked vertically.

But are they connected?

Yes — the top block is on top of the main block, so the total volume is just the sum.

But the top block is only 5 cm long, and the main block is 9 cm long — so the top block sits on the middle or end?

But since it's an L-shape, likely the top block is on one end.

But regardless, volume is additive as long as they don’t overlap in space.

Here, the top block is on top of the main block, so they share a face, but do not overlap in volume.

So:

- Main block: 9 × 2 × 3 = 54 cm³
- Top block: 5 × 1 × 3 = 15 cm³
- Total = 54 + 15 = 69 cm³

But wait — is the top block sitting on the entire main block? No — it’s only 5 cm long, so it sits on a portion.

But volume is still additive.

So ✔ Answer for Problem 3: 69 cm³

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✔ Final Answers:

1) 176 mm³
2) 352 ft³
3) 69 cm³

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📝 Summary:

To find the volume of an L-block:
1. Break it into two rectangular prisms.
2. Calculate the volume of each: $ V = l \times w \times h $
3. If they overlap, subtract the overlap volume.
4. Otherwise, add the volumes.

In these cases:
- Problem 1: Two rectangles sharing a 8×2×2 region → 160 + 48 - 32 = 176 mm³
- Problem 2: Two rectangles sharing a 4×4×4 region → 160 + 256 - 64 = 352 ft³
- Problem 3: Two rectangles stacked, no overlap → 54 + 15 = 69 cm³

Let me know if you want diagrams or further explanation!