Compound Solids Volume Worksheet - Calculate the volume of each shape.

Let’s solve each compound solid one by one. We’ll break each shape into two rectangular prisms (boxes), find the volume of each box, and then add them together.

Volume of a rectangular prism = length × width × height

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Problem 1: Top Left Shape

This is an L-shaped solid. Let’s split it vertically:

- Left part: 5 cm long, 4 cm wide, 3 cm high → Volume = 5 × 4 × 3 = 60 cm³
- Right part: It sticks out on top. The total length is 9 cm, so right part length = 9 - 5 = 4 cm. Width is still 4 cm? Wait — look again.

Actually, looking at dimensions:

The base is 9 cm long, 4 cm wide. The left side is 5 cm long and 3 cm high. The right side goes up to 5 cm high.

Better way: Split horizontally.

Bottom layer: entire base 9 cm × 4 cm × 3 cm → 9×4×3 = 108 cm³

Top layer: only on the right part. Length = 9 - 5 = 4 cm, width = 4 cm, height = 5 - 3 = 2 cm → 4×4×2 = 32 cm³

Total = 108 + 32 = 140 cm³

Wait — let me check with vertical split:

Left block: 5 cm (L) × 4 cm (W) × 3 cm (H) = 60 cm³

Right block: 4 cm (L) × 4 cm (W) × 5 cm (H) = 80 cm³

But that would be if they were separate — but in the diagram, the right block sits on top of the same base? No — actually, the whole thing is one piece. If we do vertical split, the right block should be from x=5 to x=9, which is 4 cm long, full height 5 cm, and depth 4 cm → 4×4×5 = 80 cm³

Left block: 5×4×3 = 60 cm³

Total = 60 + 80 = 140 cm³ — same as before. Good.

✔ Answer for #1: 140 cm³

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Problem 2: Top Right Shape

It looks like a big rectangle with a smaller rectangle cut out from the front? Or maybe two blocks stuck together?

Dimensions: Total length 7 m, height 5 m, depth 3 m. There’s a dashed line showing a section that’s 2 m wide and 3 m deep? Actually, looking at labels:

The figure has outer dimensions: 7 m long, 5 m high, 3 m deep.

There’s a notch or step: from the front, there’s a part that’s 2 m wide (along the length) and 3 m deep, and height? The label says “3 m” next to the inner edge — probably meaning the inner part is 3 m high? Wait, no.

Looking carefully: The solid is like a large box with a smaller box attached to the side? Or perhaps it's a single block with a recess?

Actually, better interpretation: It’s composed of two rectangular prisms joined side by side.

Left part: 5 m long? Wait, total length is 7 m. The dashed line shows a division at 2 m from the right? Label says “2 m” near the right end.

Assume:

- Right part: 2 m long, 3 m wide (depth), 5 m high → 2×3×5 = 30 m³

- Left part: remaining length = 7 - 2 = 5 m, same depth 3 m, but height? The diagram shows the left part is shorter in height? No — actually, both parts seem to have same height? Wait, no — the left part has a lower height? Look at the drawing: the left part goes down to 3 m height? But label says “3 m” on the side — maybe that’s the depth.

I think I misread. Let me re-express:

From the diagram:

Overall: length 7 m, height 5 m, depth 3 m.

There is a vertical cut at 2 m from the right end. So:

- Right prism: 2 m (length) × 3 m (depth) × 5 m (height) = 30 m³

- Left prism: 5 m (length) × 3 m (depth) × ? height. But the diagram shows the left part is only 3 m high? Wait, no — the label “3 m” is written along the depth direction, not height.

Actually, all dimensions are labeled clearly:

On the right side: “2 m” is the length of the right segment.

“3 m” is the depth (going into page).

“5 m” is the total height.

And on the left part, there’s a label “3 m” — but that might be indicating the height of the left part? No, because the left part appears to go full height.

Wait — perhaps it’s not two separate heights. Maybe the entire solid is uniform height 5 m, and it’s just divided into two sections along the length.

In that case:

Total volume without any cut: 7 × 3 × 5 = 105 m³

But that can’t be — because why show the dashed line? Probably it’s two separate blocks.

Another possibility: The solid is made of two blocks:

Block A: 5 m long × 3 m deep × 5 m high? But that doesn't match.

Look at the numbers given:

Labels: 7 m (total length), 5 m (height), 3 m (depth), and 2 m (on the right part).

Perhaps the left part is 5 m long, 3 m deep, 3 m high? And the right part is 2 m long, 3 m deep, 5 m high?

That makes sense with the drawing — the left part is shorter in height.

Yes! In the diagram, the left portion has a lower top surface. So:

- Left block: length = 7 - 2 = 5 m, depth = 3 m, height = 3 m → 5×3×3 = 45 m³

- Right block: length = 2 m, depth = 3 m, height = 5 m → 2×3×5 = 30 m³

Total = 45 + 30 = 75 m³

✔ Answer for #2: 75 m³

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Problem 3: Middle Left Shape

Trapezoidal prism? Or stepped?

Dimensions: Total length 10 cm, width 4 cm. Height on left is 3 cm, on right is 5 cm? Wait, labels:

It says: 10 cm (length), 4 cm (width/depth), and heights: left side 3 cm, right side 5 cm? But also a label “2 cm” — probably the difference.

Actually, it looks like a ramp or wedge. Can be seen as a rectangular prism plus a triangular prism? Or split into two rectangles.

Easier: Think of it as a prism with trapezoidal cross-section.

Cross-section area = average height × width = [(3 + 5)/2] × 4? No — wait, the 4 cm is the depth (into page), and the face is a trapezoid with parallel sides 3 cm and 5 cm, and distance between them is 10 cm? That doesn’t make sense.

Standard way: For a prism with trapezoidal base, volume = area of trapezoid × depth.

Here, the trapezoid is on the front face: bases 3 cm and 5 cm, height (distance between bases) is 10 cm? But that would be very flat.

No — looking at the diagram: the solid is 10 cm long (front to back?), 4 cm wide (left to right), and the height varies from 3 cm at one end to 5 cm at the other.

Actually, typically in such diagrams, the 10 cm is the length along which the height changes, and 4 cm is the constant depth.

So, cross-section perpendicular to the 10 cm direction is a rectangle whose height changes linearly.

We can compute volume as average height × length × depth.

Average height = (3 + 5)/2 = 4 cm

Length = 10 cm

Depth = 4 cm

Volume = 4 × 10 × 4 = 160 cm³

Alternatively, split into a rectangular prism and a triangular prism.

Rectangular part: 10 cm × 4 cm × 3 cm = 120 cm³

Triangular part on top: base 10 cm, height 2 cm (since 5-3=2), depth 4 cm → volume of triangular prism = (1/2 × base × height) × depth = (1/2 × 10 × 2) × 4 = 10 × 4 = 40 cm³

Total = 120 + 40 = 160 cm³

✔ Answer for #3: 160 cm³

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Problem 4: Middle Right Shape

Looks like a cube with a smaller cube cut out? Or two blocks.

Dimensions: Overall 7 m × 7 m × 7 m? But there’s a notch.

Labels: 7 m (side), 3 m (inner dimension), 4 m (another dimension).

Specifically: The solid is 7 m tall, 7 m wide, and 7 m deep? But there’s a rectangular hole or step.

From the diagram: It seems like a large cube of 7x7x7, but with a smaller rectangular prism removed from the corner.

The removed part: 3 m wide, 4 m deep, and how high? The label “3 m” is on the side, probably indicating the height of the removed part? Or the remaining?

Actually, looking: The front face has a step: from bottom, up 4 m, then over 3 m, then up to 7 m.

So, we can think of it as two blocks:

Bottom block: 7 m wide × 7 m deep × 4 m high → 7×7×4 = 196 m³

Top block: only on the left part? Width = 7 - 3 = 4 m? Depth = 7 m? Height = 7 - 4 = 3 m → 4×7×3 = 84 m³

Total = 196 + 84 = 280 m³

But is the depth full 7 m for both? Yes, assuming the step is only in width and height, not depth.

Alternative: Total volume if solid cube: 7×7×7 = 343 m³

Removed part: 3 m (width) × 7 m (depth) × 3 m (height) = 63 m³? Because from the top, we remove a block that is 3m wide, full depth 7m, and height 3m (since total height 7m, bottom is 4m, so top part removed is 3m high).

Then volume = 343 - 63 = 280 m³ — same.

✔ Answer for #4: 280 m³

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Problem 5: Bottom Left Shape

Two cubes or blocks side by side.

Left block: 8 ft × 8 ft × 8 ft? Labels: 8 ft (height), 8 ft (width), and depth? Not labeled, but probably same.

Actually, labels: left part 8 ft high, 8 ft wide; right part 6 ft high, 5 ft wide? And total depth? Assume depth is consistent.

Looking: The solid has two parts:

- Left: 8 ft (height) × 8 ft (width) × ? depth. But depth isn't labeled. However, since it's a compound solid and no depth given separately, probably the depth is the same for both and implied.

Wait, in the diagram, for the left block, it shows 8 ft height and 8 ft width, and for the right block, 6 ft height and 5 ft width. But what about depth? Probably, the depth is uniform and not specified, but in such problems, if not given, we assume the third dimension is provided or same.

Actually, looking back at other problems, depth was given. Here, perhaps the "8 ft" on the left includes depth? Unlikely.

Another thought: Perhaps the 8 ft on the left is height, and the width is 8 ft, and depth is also 8 ft? But then right part is different.

Notice: The right part has label "6 ft" height and "5 ft" width, and there's a dashed line suggesting the depth is shared.

Probably, the depth is the same for both blocks, and it's 8 ft? But not stated.

Wait, in the diagram, for the left block, it might be a cube: 8x8x8, but then the right block is attached, with width 5 ft, height 6 ft, and same depth 8 ft.

That makes sense.

So:

- Left block: 8 ft × 8 ft × 8 ft = 512 ft³

- Right block: 5 ft (width) × 8 ft (depth) × 6 ft (height) = 240 ft³

Total = 512 + 240 = 752 ft³

But is the depth really 8 ft? The problem doesn't specify, but in context, since left is shown as cube-like, and no other dimension, likely depth is 8 ft.

To confirm: if depth were different, it would be labeled. So assume depth = 8 ft for both.

✔ Answer for #5: 752 ft³

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Problem 6: Bottom Right Shape

A large rectangular prism with a smaller one on top? Or a step.

Dimensions: Total length 12 mm, width 6 mm, height: bottom part 3 mm, top part additional 2 mm? Labels: 12 mm (length), 6 mm (width), 3 mm (bottom height), 2 mm (top height increment), and 8 mm? Wait, "8 mm" is labeled on the side — probably the width of the top part.

Specifically: The solid has a base: 12 mm long, 6 mm wide, 3 mm high.

On top of it, there is a smaller block: 8 mm long, ? wide, 2 mm high.

What is the width of the top block? Probably same as base, 6 mm, unless specified.

In the diagram, it seems the top block is centered or aligned, and width is not reduced, so assume width = 6 mm.

So:

- Bottom block: 12 × 6 × 3 = 216 mm³

- Top block: 8 × 6 × 2 = 96 mm³

Total = 216 + 96 = 312 mm³

But is the top block's width really 6 mm? The label "8 mm" is on the length, and no other width given, so yes.

✔ Answer for #6: 312 mm³

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Now, compiling all answers:

1. 140 cm³
2. 75 m³
3. 160 cm³
4. 280 m³
5. 752 ft³
6. 312 mm³

Final Answer:
1. 140 cm³
2. 75 m³
3. 160 cm³
4. 280 m³
5. 752 ft³
6. 312 mm³