Geometry worksheet on volumes of prisms and cylinders with illustrated problems and calculations.

Let’s solve each problem one by one, step by step. We’ll use the volume formulas for prisms and cylinders:

- Volume of a prism = Base Area × Height
- Volume of a cylinder = π × r² × h (where r is radius, h is height)

We’ll also remember:
- For a circle: diameter = 2 × radius → so radius = diameter ÷ 2
- For area of parallelogram: base × height
- For area of triangle: (base × height) ÷ 2
- For area of trapezoid: ((base1 + base2) ÷ 2) × height

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Problem 1: Right prism with parallelogram base

Base is a parallelogram: base = 14, height = 10 → area = 14 × 10 = 140
Height of prism = 36
Volume = 140 × 36 = ?

Let’s compute:
140 × 30 = 4200
140 × 6 = 840
Total = 4200 + 840 = 5040

✔ Answer: 5040 cubic units

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Problem 2: Cylinder with diameter 18 inches, height 22

Radius = 18 ÷ 2 = 9 inches
Volume = π × r² × h = π × 9² × 22 = π × 81 × 22

Compute 81 × 22:
80 × 22 = 1760
1 × 22 = 22
Total = 1782
So volume = 1782π cubic inches

If we need decimal approximation (using π ≈ 3.14):
1782 × 3.14 = ?

Break it down:
1782 × 3 = 5346
1782 × 0.14 = let’s do 1782 × 0.1 = 178.2; 1782 × 0.04 = 71.28 → total 178.2 + 71.28 = 249.48
Add to 5346: 5346 + 249.48 = 5595.48

But since the problem doesn’t specify, leaving in terms of π is fine unless told otherwise. However, looking at other problems, they may expect numerical answers. Let’s check context — Problem 5 uses “144π”, so maybe leave as 1782π? But Problem 3 says “nearest cubic foot”, so probably wants number.

Wait — Problem 2 doesn’t say “in terms of π” or “use 3.14”. In many worksheets, if not specified, you can leave as multiple of π. But let’s see what makes sense.

Actually, looking ahead — Problem 5 gives answer in terms of π (“144π”), so likely acceptable to leave as 1782π. But to be safe, I’ll provide both and note.

However, since no instruction, and others like #3 ask for rounded, but #2 doesn’t — perhaps exact form is okay. But let me re-read: “Find the volume...” — no specification. In school, often they want numerical value using π=3.14 unless stated.

I think best to give numerical value rounded reasonably. Since inputs are whole numbers, round to nearest whole number.

So 1782 × 3.1416 ≈ let's calculate properly:

1782 × 3.14 = 5595.48 as above → rounds to 5595

But let me verify multiplication:

81 × 22 = 1782 ✔️
1782 × 3.14:

Do 1782 × 314 / 100

First, 1782 × 300 = 534600
1782 × 14 = 1782×10=17820; 1782×4=7128 → 17820+7128=24948
Total = 534600 + 24948 = 559548
Divide by 100 → 5595.48 → yes.

So approximately 5595 cubic inches

But wait — maybe they want exact? Hmm. Looking at problem 5, it says “144π”, so perhaps here too. But problem 2 doesn't have π in given data. Actually, diameter is given, so volume will involve π.

I think safest is to write 1782π, but since student might need number, and worksheet has mixed, I'll go with numerical as most common expectation unless specified.

Actually, let me check problem 8 — similar, will handle later.

For now, I’ll put 1782π as exact, but since user didn’t specify, and to match style, perhaps compute numerically. Wait — problem 3 says “to nearest cubic foot”, implying others may not require rounding. Problem 2 has no such instruction.

Looking back at image description — actually, in original, problem 2 just says “find the volume”, no mention of π or decimal. In many curricula, they accept either, but often prefer simplified exact form.

But to avoid confusion, I’ll provide the exact expression and then the approximate.

However, for consistency with final answer format, I need to pick one. Let me see problem 5: it says “volume is 144π”, so they are comfortable with π in answer. So for problem 2, answer is 1782π cubic inches.

Yes, that’s better. No need to approximate unless asked.

✔ Answer: 1782π cubic inches

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**Problem 3: Pool shaped like cylinder? Diagram shows oval but labeled 75 ft long, 18 ft wide? Wait — it says “pool”, diagram looks like elliptical cylinder? But typically in such worksheets, if it’s drawn as rectangle with semicircles, but here it’s labeled 75 ft and 18 ft — probably it’s a rectangular prism? Wait no — the diagram shows a flat shape with length 75 and width 18, and it’s called “pool”, and asks for water amount — likely it’s a rectangular prism pool? But it says “to the nearest cubic foot”, and diagram has curved ends? Actually, looking at description: “oval” but labeled 75 ft and 18 ft — probably it’s meant to be a cylinder? No, 75 ft is length, 18 ft is diameter? The diagram shows a side view with depth? Wait, the text says “amount of water in the pool”, and diagram has dimensions 75 ft and 18 ft — likely it’s a rectangular pool? But it’s drawn with rounded ends.

Actually, re-examining: in many such problems, if it’s a swimming pool with straight sides and semi-circular ends, it’s a combination, but here only two dimensions given: 75 ft and 18 ft. And it’s labeled “18 ft” on the short side, “75 ft” on long side. Probably, it’s intended to be a rectangular prism? But why “oval” appearance? Perhaps it’s a cylinder lying on its side? That would make sense: length 75 ft, diameter 18 ft.

Yes! That must be it. Because if it were rectangular, it would be straightforward, but the drawing suggests circular cross-section. So assume it’s a cylindrical tank lying horizontally: length = 75 ft (which is height of cylinder), diameter = 18 ft → radius = 9 ft.

Volume = π r² h = π × 9² × 75 = π × 81 × 75

Compute 81 × 75:
80 × 75 = 6000
1 × 75 = 75
Total = 6075
So volume = 6075π cubic feet

Now, “to the nearest cubic foot” — so use π ≈ 3.1416

6075 × 3.1416 ≈ ?

First, 6000 × 3.1416 = 18849.6
75 × 3.1416 = 235.62
Total ≈ 18849.6 + 235.62 = 19085.22 → rounds to 19085

But let me compute exactly:

6075 × 3.14 = 6075 × 3 = 18225; 6075 × 0.14 = 850.5; total 19075.5
Then 6075 × 0.0016 = about 9.72, so total approx 19085.22 — yes.

So 19085 cubic feet

✔ Answer: 19085

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Problem 4: Right triangular prism

Base is triangle: base = 15, height = 12 → area = (15 × 12)/2 = 180/2 = 90
Height of prism = 20
Volume = 90 × 20 = 1800

✔ Answer: 1800 cubic units

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Problem 5: Right cylinder, volume = 144π, height = 4, find total surface area

Volume = π r² h = 144π
So π r² * 4 = 144π
Divide both sides by π: 4 r² = 144
r² = 36 → r = 6

Total surface area of cylinder = 2πr² + 2πrh
= 2π(6)² + 2π(6)(4)
= 2π(36) + 2π(24)
= 72π + 48π = 120π

✔ Answer: 120π square inches

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Problem 6: Right prism with hexagonal base? Diagram shows hexagon, side 6 feet, height 12 feet

It’s a regular hexagonal prism? Side length 6 ft, height 12 ft.

Area of regular hexagon = (3√3 / 2) × s², where s is side length.

s = 6
s² = 36
Area = (3√3 / 2) × 36 = (3×36 / 2) √3 = (108 / 2) √3 = 54√3

Volume = base area × height = 54√3 × 12 = 648√3 cubic feet

But is this expected? Maybe they consider it as composed of triangles.

Regular hexagon can be divided into 6 equilateral triangles.

Each equilateral triangle with side 6: area = (√3 / 4) × s² = (√3 / 4) × 36 = 9√3

Six of them: 6 × 9√3 = 54√3 — same as above.

So volume = 54√3 × 12 = 648√3

But perhaps they want numerical value? The problem doesn’t specify. Looking at others, some have π, some don’t. Here, no π, but irrational.

In many school contexts, they might expect the exact form or approximate.

But since no instruction, and to be precise, I’ll leave as 648√3.

However, let me check if the base is indeed regular hexagon — diagram shows symmetric, so yes.

Alternatively, sometimes they approximate √3 ≈ 1.732, but again, not specified.

I think for accuracy, keep exact.

But let me see problem 7 — has trapezoid, which may require calculation.

For now, ✔ Answer: 648√3 cubic feet

But wait — perhaps I misread. The diagram might not be regular? It shows all sides equal, angles equal, so regular.

Another way: sometimes in worksheets, they give apothem or something, but here only side given.

I think 648√3 is correct.

But to confirm, let's compute numerical: √3 ≈ 1.732, 648 × 1.732 ≈ ?

600 × 1.732 = 1039.2
48 × 1.732 ≈ 48 × 1.7 = 81.6; 48 × 0.032 = 1.536; total 83.136
Sum ≈ 1039.2 + 83.136 = 1122.336 — but since not asked, better leave exact.

However, looking back at problem 3, they asked for nearest, here no such thing. So exact is fine.

But let me double-check the figure: it says "right prism" and shows hexagon with side 6, height 12. Yes.

Perhaps they consider it as two trapezoids or something, but no, standard is regular hexagon.

I'll go with 648√3.

But wait — in some curricula, they might expect the formula without radical, but I think it's acceptable.

Alternatively, maybe the base is not regular? But diagram suggests it is.

Another thought: perhaps it's a hexagonal prism but not regular? But no indication.

I think it's fine.

✔ Answer: 648√3 cubic feet

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Problem 7: Right prism with isosceles trapezoid base

Trapezoid bases: top = 16, bottom = 42, legs = 20 each, height of trapezoid? Not given directly.

Need to find height of trapezoid first.

Isosceles trapezoid: difference in bases = 42 - 16 = 26, so each overhang is 26/2 = 13 on each side.

Then, leg = 20, overhang = 13, so height h satisfies: h² + 13² = 20²
h² + 169 = 400
h² = 400 - 169 = 231
h = √231

But √231 is messy. Did I misread?

Diagram shows: top base 16, bottom 42, legs 20, and height of prism is 30.

But to find area of trapezoid, need height of trapezoid.

With legs 20, and horizontal projection 13, yes, h = √(20² - 13²) = √(400 - 169) = √231

√231 = √(3×7×11) — no simplification.

Then area of trapezoid = ((b1 + b2)/2) × h = ((16 + 42)/2) × √231 = (58/2) × √231 = 29√231

Then volume = base area × height of prism = 29√231 × 30 = 870√231

This seems very messy for a worksheet. Probably I made a mistake.

Perhaps the 20 is not the leg, but something else? Diagram shows arrows on the slanted sides labeled 20, so likely legs.

Maybe the height is given? No.

Another possibility: perhaps the trapezoid height is implied or can be found differently.

Or maybe it's not isosceles? But it says "isosceles trapezoid".

Perhaps the 20 is the height? But no, it's labeled on the slant.

Let me read the problem: "its base is an isosceles trapezoid" and diagram has top 16, bottom 42, sides 20, and prism height 30.

Perhaps in some contexts, they expect to use Pythagorean theorem as I did.

But √231 is approximately 15.2, then area = 29 * 15.2 ≈ 440.8, times 30 ≈ 13224, but not nice.

Maybe I miscalculated the overhang.

Bases 16 and 42, difference 26, so when you drop perpendiculars from ends of top base to bottom, you get two right triangles on sides, each with base (42-16)/2 = 13, hypotenuse 20, so height h = sqrt(20^2 - 13^2) = sqrt(400-169)=sqrt(231), yes.

Perhaps the 20 is the height of the trapezoid? But the diagram shows it on the slant side.

Looking back at user's description: "diagram shows trapezoid with top 16, bottom 42, sides 20, and prism height 30" — and "isosceles trapezoid".

Perhaps in the actual image, the 20 is labeled as the height, but according to text, it's on the side.

To resolve, let's assume it's correct, and proceed.

But for school level, they might have intended different numbers. Perhaps the leg is 25 or something, but it's given as 20.

Another thought: maybe the 20 is the length of the non-parallel side, and we have to live with it.

So volume = 870√231 cubic units.

But let's see if 231 can be simplified — 231 = 3*77 = 3*7*11, no square factors, so yes.

Perhaps they want numerical value. Let's calculate.

√231 ≈ 15.1987
29 * 15.1987 ≈ 29*15 = 435, 29*0.1987≈5.7623, total 440.7623
Times 30 = 13222.869 — rounds to 13223

But again, not specified.

Perhaps I misinterpreted the diagram. Another possibility: maybe the 20 is the height of the trapezoid, not the leg. But the problem says "isosceles trapezoid" and typically legs are equal, and if 20 were height, it would be labeled vertically.

In the user's description, it says "diagram shows ... sides 20", so likely legs.

To move forward, I'll assume it's correct, and provide exact form.

But let's check problem 8 for comparison.

Perhaps for this worksheet, they expect us to calculate the height.

So h = sqrt(20^2 - ((42-16)/2)^2) = sqrt(400 - 13^2) = sqrt(400-169) = sqrt(231)

Then area = (16+42)/2 * sqrt(231) = 29 sqrt(231)

Volume = 29 sqrt(231) * 30 = 870 sqrt(231)

I think that's it.

✔ Answer: 870√231 cubic units

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Problem 8: Cylindrical can, soup volume — can is tilted? Diagram shows can pouring, but says "cylindrical can that is 8 inches tall and 4 inches across the lid"

"Across the lid" means diameter = 4 inches, so radius = 2 inches.

Height = 8 inches.

But is the can full? The diagram shows it pouring, but the problem says "find the volume of soup", implying how much is in it, but no information on how much is poured out.

Re-read: "Find the volume of soup in a cylindrical can that is 8 inches tall and 4 inches across the lid."

And diagram shows a can tilted, pouring liquid, but no indication of how much is left.

This is ambiguous. Probably, it's assuming the can is full, and we need to find its capacity.

Because if it's pouring, we don't know how much is left.

In many such problems, "volume of soup in the can" means the capacity, especially if no other info.

Moreover, the diagram might be illustrative.

So likely, it's a full cylinder: radius = 4/2 = 2 inches, height = 8 inches.

Volume = π r² h = π × 4 × 8 = 32π cubic inches.

If they want numerical, 32 × 3.14 = 100.48, but again, not specified.

Given problem 2 and 5 have π, likely 32π is fine.

But let's see the context — problem 3 asked for nearest, here no.

So I'll go with 32π.

✔ Answer: 32π cubic inches

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Now, compiling all answers:

1. 5040
2. 1782π
3. 19085
4. 1800
5. 120π
6. 648√3
7. 870√231
8. 32π

But for problem 6 and 7, the radicals might be unexpected. Let me double-check problem 6.

Problem 6: hexagonal prism, side 6 ft, height 12 ft.

Area of regular hexagon = (3√3/2) * s^2 = (3√3/2)*36 = 54√3, yes.

Volume = 54√3 * 12 = 648√3, correct.

For problem 7, perhaps the 20 is the height of the trapezoid? But the problem says "isosceles trapezoid" and labels the sides as 20, which are the legs.

Maybe in the diagram, the 20 is the height. Let me assume that for sanity.

Suppose the height of the trapezoid is 20. Then area = ((16+42)/2) * 20 = 29 * 20 = 580

Volume = 580 * 30 = 17400

That's nice number. And 20 is labeled on the side, but perhaps it's a mistake in interpretation.

In many textbooks, when they label the slant side, they intend for you to calculate height, but here it results in ugly number.

Perhaps the legs are 25 or 26, but it's given as 20.

Another idea: perhaps "20" is the length of the leg, but in the diagram, it's shown as the height? Unlikely.

To match typical worksheet difficulty, probably they meant the height of the trapezoid is given, but it's not.

Let's look at the user's description: "diagram shows trapezoid with top 16, bottom 42, sides 20, and prism height 30"

And "isosceles trapezoid".

Perhaps the 20 is the height. I recall that in some diagrams, they label the height on the side by mistake, but usually not.

Maybe the difference is not 26. Bases 16 and 42, difference 26, half is 13, leg 20, so height sqrt(400-169)=sqrt(231)≈15.2, as before.

Perhaps they want us to leave it, or perhaps I need to box the answer as is.

For the sake of completing, I'll keep it as 870√231, but suspect it might be a typo, and perhaps the leg is 25 or something.

If leg were 25, then h = sqrt(625 - 169) = sqrt(456) = 2sqrt(114), still messy.

If leg were 26, h = sqrt(676 - 169) = sqrt(507) = sqrt(3*13^2) = 13sqrt(3), then area = 29 * 13sqrt(3) = 377sqrt(3), volume = 377sqrt(3)*30 = 11310sqrt(3), still messy.

If the height of trapezoid is 15, then area = 29*15=435, volume=13050, nice, but not given.

Perhaps the 20 is the height. Let me assume that for practical purposes, as otherwise it's too hard.

In many online sources, similar problems have the height given or calculable nicely.

Perhaps "20" is the length of the non-parallel side, and we have to use it, so I'll stick with 870√231.

But to provide a clean answer, and since the worksheet might expect numerical, but not specified.

Another thought: in problem 7, the prism height is 30, and trapezoid has bases 16, 42, legs 20, so perhaps calculate height as above.

I think for accuracy, I'll keep it.

So final answers:

1. 5040
2. 1782π
3. 19085
4. 1800
5. 120π
6. 648√3
7. 870√231
8. 32π

Now for the Final Answer section, since there are multiple problems, I need to list them all.

The user said "solve the problem", but there are 8 problems, so I should provide all answers.

In the response, I'll list them clearly.

Also, for problem 2, I decided on 1782π, but earlier I thought about numerical, but since problem 5 has π, it's consistent.

Similarly for 8.

For problem 3, I have 19085, which is numerical.

For 6 and 7, radicals.

Perhaps for 6, they expect the numerical value, but I think exact is better.

I think it's fine.

So let's write the final answers.