Worksheet on calculating the volume of prisms and cylinders with various geometric shapes and dimensions.

Let’s solve each problem step by step. We’ll calculate area of base and volume for prisms and cylinders, and find missing dimensions when given volume.

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Problem 1: Rectangular Prism

Dimensions:
Length = 14.5 m
Width = 3 m
Height = 5 m

→ Area of Base = length × width = 14.5 × 3
Calculate:
14.5 × 3 = (14 × 3) + (0.5 × 3) = 42 + 1.5 = 43.5 m²

→ Volume = Area of Base × Height = 43.5 × 5
43.5 × 5 = (40 × 5) + (3.5 × 5) = 200 + 17.5 = 217.5 m³

✔ Check: 14.5 × 3 × 5 = 14.5 × 15 = 217.5 → Correct.

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Problem 2: Triangular Prism

Base is a triangle with:
- Base of triangle = 8 m
- Height of triangle = 6 m (perpendicular height shown)
- Length of prism (distance between triangular bases) = 12 m

Wait — actually, looking at the diagram description:
The triangular face has sides labeled 8m, 8m, and base? But there's a perpendicular height of 6m drawn to the base of 8m? Actually, in the diagram, it shows:

Triangle base = 8 m (the side with the right angle mark), height = 6 m → so area of triangle = (1/2)×base×height = (1/2)×8×6 = 24 m²

Then the prism extends 12 m long? Wait — no, look again: The figure shows a triangular prism where the triangular face has base 8m, height 6m, and the “length” of the prism (the distance between the two triangular faces) is 12 m? But also there’s an 8m label on the slanted edge? Hmm.

Actually, re-examining standard interpretation: In such diagrams, if a triangle has base 8m and height 6m (with right angle marked), then area of base (triangle) = ½ × 8 × 6 = 24 m².

Then the prism’s length (the dimension going into the page or along the rectangle) is 12 m? But wait — the diagram also labels one rectangular face as 8m by 12m? That might be confusing.

Wait — let me clarify based on typical worksheet problems:

In Problem 2, the triangular base has:
- Base = 8 m
- Height = 6 m (given with perpendicular symbol)
→ So Area of Base = ½ × 8 × 6 = 24 m²

Then the prism’s length (the third dimension, perpendicular to the triangular base) is 12 m? But the diagram also says “8 m” on another edge — that might be the hypotenuse of the triangle? Let’s check: If base=8, height=6, then hypotenuse = √(8²+6²)=√(64+36)=√100=10 — but diagram says 8m? That doesn’t match.

Wait — perhaps I misread. Looking again: The diagram likely shows a triangular prism where the triangular face has sides 8m, 8m, and base? No — better to go by what’s labeled with perpendicular height.

Actually, common setup: Triangle with base 8m, height 6m → area = 24 m². Then the prism extends 12 m in the direction perpendicular to the triangle → so volume = 24 × 12 = 288 m³.

But wait — the diagram also labels “5.5 m” inside? And “8 m” on top? This is ambiguous without seeing exact layout.

Alternative interpretation: Maybe the triangular base has base = 8 m, height = 6 m → area = 24 m². The length of the prism is 5.5 m? Because sometimes the “length” is the dimension connecting the two triangles.

Looking at the labels: It says “8 m” on the left edge of triangle, “6 m” as height, “12 m” as the bottom edge of the rectangle? And “5.5 m” as the depth?

Actually, let’s assume standard labeling: For a triangular prism, volume = (area of triangular base) × (length of prism).

If the triangle has base 8 m and height 6 m → area = 24 m².

Now, which dimension is the “length” of the prism? Typically, it’s the dimension perpendicular to the triangular face. In many worksheets, if they show a rectangle attached to the triangle with length 12 m, that’s the prism length.

But here, there’s also “5.5 m” labeled — possibly that’s the length? Or maybe 12 m is the length?

Wait — let’s think differently. Perhaps the triangular face is not the one with base 8 and height 6? Maybe the 8m and 8m are the equal sides, and 6m is the height to the base? Then base would be calculated via Pythagoras: half-base = √(8² - 6²) = √(64-36)=√28≈5.29 — messy.

This is getting too complicated. Let me use the most straightforward reading:

In many such worksheets, when they draw a triangle with base labeled and height labeled with perpendicular symbol, you use those for area.

So: Triangle base = 8 m, height = 6 m → Area = ½ * 8 * 6 = 24 m²

Then the prism’s length (the dimension extending out) is 12 m? But why is 5.5 m there? Maybe 5.5 m is the length? Let’s check both.

Option A: Length = 12 m → Volume = 24 * 12 = 288 m³

Option B: Length = 5.5 m → Volume = 24 * 5.5 = 132 m³

Which makes more sense? Also, there’s an “8 m” labeled on the top edge — perhaps that’s the length? I’m confused.

Wait — let’s look at the numbers: 8, 6, 12, 5.5. Perhaps the triangular base has sides 8, 8, and base unknown, but height 6 to the base. Then base = 2 * sqrt(8^2 - 6^2) = 2*sqrt(64-36)=2*sqrt(28)=2*2*sqrt(7)=4sqrt(7)≈10.58 — not nice.

Alternatively, maybe the 6m is not the height of the triangle but something else? No, it has a right angle symbol.

Another idea: Perhaps the "8 m" on the left is the height of the triangle, and the base is 6 m? But it's labeled with 6m as the vertical leg.

I think the intended interpretation is:

Triangular base: base = 8 m, height = 6 m → area = 24 m²

Prism length = 12 m → volume = 24 * 12 = 288 m³

And the 5.5 m might be a red herring or for another part? Or perhaps it's the length? Let's see the answer choices later.

Wait — in some diagrams, the 5.5 m could be the length of the prism. Let me calculate both and see which gives integer or nice number.

24 * 5.5 = 132 — nice number.

24 * 12 = 288 — also nice.

But 5.5 is decimal, 12 is integer. Usually worksheets use integers.

Perhaps the 12 m is the base of the rectangle, and 5.5 m is the length of the prism.

Let's assume that the triangular face has area 24 m², and the prism extends 5.5 m in the direction perpendicular to the triangle. Why? Because in the diagram, the 5.5 m is likely the "depth" or length of the prism, while 12 m is part of the rectangular face.

Actually, upon second thought, in a triangular prism, the volume is area of triangle times the length between the two triangular faces. If the diagram shows a rectangle with dimensions 12 m by 8 m, that might be one lateral face, but the length of the prism is the dimension perpendicular to the triangle.

I recall that in some textbooks, for a triangular prism, if they give the three dimensions of the rectangular faces, but here it's mixed.

Let's try this: Suppose the triangular base has base b and height h, area = ½bh.

From the diagram, we have a right triangle with legs 6 m and ? , hypotenuse 8 m? Then other leg = sqrt(8^2 - 6^2) = sqrt(64-36) = sqrt(28) = 2sqrt(7) ≈5.29 — not good.

Perhaps the 8 m is the base, 6 m is the height, and the 12 m is the length of the prism. I think that's the most reasonable assumption.

Moreover, in problem 4, they have similar setup with inches, and it's clear.

For consistency, let's go with:

Area of Base (triangle) = ½ * 8 * 6 = 24 m²

Volume = 24 * 12 = 288 m³ [assuming 12 m is the length of the prism]

But why is 5.5 m there? Perhaps it's a typo or for another purpose. Maybe the 5.5 m is the length, and 12 m is something else.

Let's read the labels carefully from the user's description: "8 m" on the left edge of the triangle, "6 m" as the height (with right angle), "12 m" as the bottom edge of the rectangle, and "5.5 m" as the depth or the length of the prism.

In many online sources, for such a diagram, the 5.5 m is the length of the prism. For example, if the triangular face is on the front, then the prism extends back 5.5 m.

So let's assume:

Area of Base = ½ * 8 * 6 = 24 m²

Length of prism = 5.5 m

Volume = 24 * 5.5 = let's calculate: 24 * 5 = 120, 24 * 0.5 = 12, total 132 m³

That seems reasonable.

To confirm, 24 * 5.5 = 132.

I think that's it. The 12 m might be the hypotenuse or something, but since we have base and height for the triangle, we don't need it.

So for Problem 2:

Area of Base = 24 m²

Volume = 132 m³

✔ Double-check: ½ * 8 * 6 = 24; 24 * 5.5 = 132 — yes.

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Problem 3: Cylinder

Diameter = 12 ft → radius r = 6 ft

Height h = 9 ft

Area of Base = πr² = π * 6² = 36π ft²

We can leave in terms of π or use 3.14. Since no specification, probably use 3.14 for calculation.

But in worksheets, sometimes they want exact or approximate. Let's see the context. In problem 6, they give volume with decimals, so likely use π ≈ 3.14.

So Area of Base = 3.14 * 36 = let's compute: 3*36=108, 0.14*36=5.04, total 113.04 ft²

Volume = Area of Base × Height = 113.04 * 9

Calculate: 113.04 * 9 = 113.04 * 10 - 113.04 = 1130.4 - 113.04 = 1017.36 ft³

Or directly: 100*9=900, 13*9=117, 0.04*9=0.36, total 900+117=1017 +0.36=1017.36 ft³

Using exact π: Volume = π * 6² * 9 = π * 36 * 9 = 324π ft³ ≈ 324 * 3.14 = 1017.36 — same.

So:

Area of Base = 113.04 ft² (if using 3.14)

Volume = 1017.36 ft³

But perhaps they want it in terms of π? The problem doesn't specify. In many middle school worksheets, they use 3.14.

Since problem 6 has decimal volume, likely use 3.14.

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Problem 4: Rectangular Prism (but tilted?)

Dimensions: 3 in, 5 in, 18 in

It's a rectangular prism, so any face can be base.

Typically, base is the bottom face. Here, it shows 3 in and 5 in as the base dimensions, and 18 in as height? Or vice versa.

The diagram shows: 3 in and 5 in on the front face, and 18 in as the length going back.

So Area of Base = 3 * 5 = 15 in²

Volume = 15 * 18 = 270 in³

Check: 3*5*18 = 15*18=270 — yes.

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Problem 5: Find missing dimension x

Figure is a rectangular prism with dimensions: 1.5 ft, x ft, 15 ft

Volume = 135 ft³

Volume = l * w * h = 1.5 * x * 15 = 135

Compute: 1.5 * 15 = 22.5

So 22.5 * x = 135

x = 135 / 22.5

Calculate: 22.5 * 6 = 135 → because 20*6=120, 2.5*6=15, total 135

So x = 6 ft

✔ Check: 1.5 * 6 * 15 = (1.5*15)*6 = 22.5*6=135 — correct.

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Problem 6: Cylinder, find radius x

Volume = 4069.44 ft³

Height = 16 ft

Radius = x ft

Volume of cylinder = π r² h

So π * x² * 16 = 4069.44

Use π ≈ 3.14

So 3.14 * x² * 16 = 4069.44

First, 3.14 * 16 = ?

3 * 16 = 48

0.14 * 16 = 2.24

Total 50.24

So 50.24 * x² = 4069.44

x² = 4069.44 / 50.24

Let me divide:

First, note that 50.24 * 80 = 50.24 * 100 - 50.24 * 20 = 5024 - 1004.8 = 4019.2

4069.44 - 4019.2 = 50.24

50.24 * 1 = 50.24

So 80 + 1 = 81

Thus x² = 81

x = 9 ft (since radius positive)

✔ Check: π * 9² * 16 = 3.14 * 81 * 16

81 * 16 = 1296

1296 * 3.14 = 1296 * 3 = 3888, 1296 * 0.14 = 181.44, total 3888 + 181.44 = 4069.44 — perfect.

So x = 9 ft

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Now, summarize all answers:

Problem 1:
Area of Base: 43.5 m²
Volume: 217.5 m³

Problem 2:
Area of Base: 24 m²
Volume: 132 m³ (using length 5.5 m)

Problem 3:
Area of Base: 113.04 ft² (using π=3.14)
Volume: 1017.36 ft³

Problem 4:
Area of Base: 15 in²
Volume: 270 in³

Problem 5:
Missing dimension x = 6 ft

Problem 6:
Missing dimension x = 9 ft

Note: For Problem 2, if the intended length was 12 m, volume would be 288 m³, but given the presence of 5.5 m and typical worksheet design, 5.5 m is likely the prism length. However, to be thorough, let's consider if 12 m is the length.

If length = 12 m, volume = 24 * 12 = 288 m³

But then what is 5.5 m? Perhaps it's irrelevant or a distractor. In some diagrams, 5.5 m might be the height of something else, but unlikely.

Given that in problem 4, they have clear dimensions, and here 5.5 m is labeled similarly to how 18 in is labeled in problem 4 (as the length), I think 5.5 m is the length.

Moreover, 24 * 5.5 = 132 is a clean number, while 24*12=288 is also clean, but 5.5 is specified.

I'll stick with 132 m³ for volume in problem 2.

Final Answers:

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Final Answer:
1.) Area of Base: 43.5 m², Volume: 217.5 m³
2.) Area of Base: 24 m², Volume: 132 m³
3.) Area of Base: 113.04 ft², Volume: 1017.36 ft³
4.) Area of Base: 15 in², Volume: 270 in³
5.) Missing dimension: 6 ft
6.) Missing dimension: 9 ft