Volume of Composite Solids worksheet featuring nine geometric figures with dimensions for calculating volume.

Let's solve three of the problems from the worksheet, making sure to include Problem #5, as required.

We’ll solve:
- Problem 1
- Problem 5
- Problem 9

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🔷 Problem 1: Volume of an L-shaped solid (in³)

This is a composite solid made of two rectangular prisms.

#### Step 1: Break into two rectangles

We can split this L-shape into:
- Vertical part: 10 in × 3 in × 8 in
- Horizontal part: 4 in × 3 in × 8 in
(Note: The horizontal base is 12 in long, but only 4 in wide because 8 in is already covered by vertical part.)

Wait — let’s look carefully at the figure:

The full base is 12 in, and the vertical section is 8 in long, so the horizontal extension is:
- Horizontal part: 12 in – 8 in = 4 in wide?

But actually, looking at the drawing:
- The vertical rectangle is 10 in tall, 8 in deep, 3 in wide
- The horizontal rectangle extends out 12 in long, but only 4 in wide, and 3 in high

Wait — actually, both parts are 3 in high, and the depth is 3 in for both.

So:
- Vertical prism: 10 in (height) × 8 in (length) × 3 in (width)
- Horizontal prism: 4 in (height) × 4 in (length) × 3 in (width)? Wait — no.

Wait! Let's re-analyze.

Actually, the entire object has:
- A tall vertical block: 10 in tall, 8 in long, 3 in wide
- A shorter horizontal block: 4 in tall, 4 in long, 3 in wide? No.

Wait — better: the total length is 12 in, and the vertical part is 8 in long, so the horizontal part must be 12 - 8 = 4 in long.

But the height of the horizontal part is 3 in, and the vertical part is 10 in, so the horizontal part is only 3 in high, while the vertical part goes up to 10 in.

So we have:
- Top part: 10 in × 8 in × 3 in
- Bottom part: 4 in × 4 in × 3 in? Wait — no. The horizontal base is 12 in long, but only 3 in high, and it runs under the entire length.

Actually, the bottom horizontal part is 12 in long, 3 in high, and 3 in wide.

But wait — the vertical part is 8 in long, 10 in high, and 3 in wide, and sits on top of the horizontal part.

So the horizontal part is:
- Length: 12 in
- Height: 3 in
- Width: 3 in

And the vertical part is:
- Length: 8 in
- Height: 7 in (since total height is 10 in, and bottom is 3 in, so upper part is 7 in)
- Width: 3 in

Wait — no! The vertical part is 10 in tall, and the horizontal part is only 3 in tall, so the vertical part is sitting on top of the horizontal one, meaning the vertical part has height 10 in, and the horizontal part is 3 in tall.

But then how does it fit?

Let’s assume the horizontal base is 12 in long, 3 in high, 3 in wide.

Then the vertical block sits on top of it, extending 8 in long, 10 in high, 3 in wide.

But the vertical block must be placed so that its base matches the horizontal block.

So:
- Volume of horizontal base = length × width × height = 12 × 3 × 3 = 108 in³
- Volume of vertical part = 8 × 3 × 10 = 240 in³

Wait — but the vertical part is only 8 in long, and the horizontal part is 12 in long, so they don't overlap perfectly.

But if the vertical part is placed on top of the horizontal part, and the horizontal part is 12 in long, and the vertical part is 8 in long, then the overlap is 8 in.

But since the vertical part is taller, it adds extra volume.

So total volume = volume of horizontal base + volume of vertical column.

But the vertical column is 8 in long, 3 in wide, 10 in high, so volume = 8 × 3 × 10 = 240 in³

The horizontal base is 12 in long, 3 in wide, 3 in high → volume = 12 × 3 × 3 = 108 in³

But now — do they overlap? Yes. The vertical column sits on top of the horizontal base, so the base is shared.

So total volume = horizontal base + vertical column = 108 + 240 = 348 in³

But wait — the vertical column is only 8 in long, so the horizontal base has 4 in extending beyond the vertical part.

So yes, the horizontal base is 12 in long, 3 in high, 3 in wide → 12×3×3 = 108

The vertical column is 8 in long, 3 in wide, 10 in high → 8×3×10 = 240

But the vertical column sits on top of the first 8 in of the horizontal base, so the total volume is:

> Total = 108 (horizontal) + (240 - 8×3×3) ??? No — we cannot subtract.

No — the horizontal base includes the bottom 3 in of the vertical column.

So when we add them, we're double-counting the overlapping region: 8 in × 3 in × 3 in = 72 in³

So total volume = horizontal + vertical − overlap
= 108 + 240 − 72 = 276 in³

Alternatively, think of it as:
- The horizontal base is 12×3×3 = 108
- The vertical extension above the base is 8×3×(10−3) = 8×3×7 = 168
- So total = 108 + 168 = 276 in³

✔ Answer: 276 in³

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🔷 Problem 5: Cylinder on top of a cube (in³)

We have:
- A cube with side 5 in
- A cylinder on top with diameter 4 in, height 4 in

Use π = 3.14

#### Step 1: Volume of cube
V_cube = s³ = 5³ = 125 in³

#### Step 2: Volume of cylinder
V_cylinder = π × r² × h
r = 4/2 = 2 in
h = 4 in
V = 3.14 × (2)² × 4 = 3.14 × 4 × 4 = 3.14 × 16 = 50.24 in³

#### Step 3: Total volume
Total = 125 + 50.24 = 175.24 in³

✔ Answer: 175.24 in³

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🔷 Problem 9: T-shaped solid (in³)

This is a T-shaped block made of two rectangular prisms.

Break it into:
- Base rectangle: 4 in long, 2 in wide, 1 in high
- Top rectangle: 1 in long, 1 in wide, 1 in high

But wait — the top piece is centered on the base.

From the diagram:
- The base is 4 in long, 2 in wide, 1 in high → volume = 4 × 2 × 1 = 8 in³
- The top is 1 in long, 1 in wide, 1 in high → volume = 1 × 1 × 1 = 1 in³

But the top is centered, so it doesn’t extend beyond the base.

So total volume = 8 + 1 = 9 in³

Wait — but the height of the entire T is 3 in, and the base is 1 in high, so the top is 2 in high?

No — the base is 1 in high, and the top is another 1 in high, so total height is 2 in, but the diagram says 3 in?

Wait — label says 3 in total height.

Look: the base is 1 in high, and the top is 2 in high? But it shows 1 in for the top.

Wait — the T shape has:
- Bottom rectangle: 4 in long, 2 in wide, 1 in high
- Top rectangle: 1 in long, 1 in wide, 1 in high — but placed on top, so total height is 1 + 1 = 2 in

But the diagram says 3 in total height? That contradicts.

Wait — maybe the top is 2 in high?

No — the labels say:
- Base: 1 in high
- Top: 1 in high
- Total height: 3 in

That doesn’t add up.

Wait — perhaps the top is 2 in high? But labeled as 1 in.

Wait — look again:

The diagram shows:
- A base of 4 in long, 2 in wide, 1 in high
- A vertical stem of 1 in wide, 1 in deep, 2 in high?
But it's labeled as 1 in for the top.

Wait — the T shape has:
- A horizontal base: 4 in long, 2 in wide, 1 in high
- A vertical stem: 1 in long, 1 in wide, 2 in high? But it’s drawn as 1 in high.

Wait — the total height is labeled as 3 in, and the base is 1 in high, so the stem is 2 in high.

But the stem is shown as 1 in wide, 1 in deep, and 2 in high.

But the label says "1 in" next to the top — probably indicating depth or width.

Let’s interpret:

- The base is: 4 in (length), 2 in (width), 1 in (height)
- The stem is: 1 in (length), 1 in (width), 2 in (height) — but it’s placed on top of the base, so it extends upward.

Wait — but the stem is 1 in wide, and the base is 2 in wide, so it fits.

But the stem is 1 in long, and the base is 4 in long, so it’s centered.

Now, volume:
- Base: 4 × 2 × 1 = 8 in³
- Stem: 1 × 1 × 2 = 2 in³

But the stem is placed on top of the base, so no overlap in volume.

So total volume = 8 + 2 = 10 in³

But the label says 1 in for the stem — which might be depth, not height.

Wait — the stem is 1 in wide, 1 in deep, 2 in high — but labeled as 1 in? Maybe it's a typo.

Wait — the diagram says:
- The total height is 3 in
- The base is 1 in high
- So the stem is 2 in high

And the stem is 1 in wide, 1 in deep, so volume = 1 × 1 × 2 = 2 in³

Base: 4 × 2 × 1 = 8 in³

Total = 8 + 2 = 10 in³

But the stem is 1 in wide, and the base is 2 in wide, so it fits.

✔ Answer: 10 in³

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✔ Final Answers:

| Problem | Volume |
|--------|--------|
| 1 | 276 in³ |
| 5 | 175.24 in³ |
| 9 | 10 in³ |

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📝 Summary of Steps:

- Problem 1: Split into base and vertical extension; avoid double-counting.
- Problem 5: Add cube and cylinder volumes using π = 3.14.
- Problem 9: Break into base and stem; sum volumes.

Let me know if you'd like solutions to other problems too!