Pyramid and Prism | PDF | Volume | Geometric Measurement - Free Printable

Let’s solve each problem one by one. We’ll calculate the volume for each shape using the correct formula.

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Problem 1: Pyramid with square base

Base side = 12 cm
Height = 15 cm

Volume of pyramid = (1/3) × Base Area × Height
Base area = 12 × 12 = 144 cm²
Volume = (1/3) × 144 × 15 = 48 × 15 = 720 cm³

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Problem 2: Rectangular prism

Length = 15 cm, Width = 6 cm, Height = 8 cm

Volume of prism = Length × Width × Height
= 15 × 6 × 8 = 90 × 8 = 720 cm³

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Problem 3: Pyramid with rectangular base

Base: 14 m × 10 m → Area = 140 m²
Height = 12 m

Volume = (1/3) × 140 × 12 = 140 × 4 = 560 m³

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Problem 4: Trapezoidal prism

This is a prism with trapezoid bases.

Trapezoid area = (1/2) × (sum of parallel sides) × height of trapezoid
Parallel sides: 12 cm and 6 cm
Height of trapezoid = 8 cm
→ Area = (1/2) × (12 + 6) × 8 = (1/2) × 18 × 8 = 9 × 8 = 72 cm²

Prism length (depth) = 15 cm
Volume = Base area × length = 72 × 15 = 1080 cm³

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Problem 5: Square pyramid

Base side = 12 in → Area = 12 × 12 = 144 in²
Height = 10 in

Volume = (1/3) × 144 × 10 = 48 × 10 = 480 in³

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Problem 6: Triangular pyramid (tetrahedron)

Base triangle: base = 12 ft, height = 10 ft → Area = (1/2) × 12 × 10 = 60 ft²
Pyramid height = 14 ft

Volume = (1/3) × 60 × 14 = 20 × 14 = 280 ft³

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Problem 7: Hexagonal prism

Regular hexagon side = 6 cm
Area of regular hexagon = (3√3 / 2) × side²
≈ (3 × 1.732 / 2) × 36 ≈ (5.196 / 2) × 36 ≈ 2.598 × 36 ≈ 93.528 cm²

But wait — looking at diagram, it shows apothem? Actually, no — let me check again.

Actually, in many school problems, if not specified, they may expect you to use given dimensions directly. But here, only side = 6 cm is given, and height of prism = 12 cm.

Wait — perhaps we can assume it's made of 6 equilateral triangles?

Area of one equilateral triangle with side 6: (√3/4) × 6² = (1.732/4) × 36 ≈ 0.433 × 36 ≈ 15.588 cm²
Hexagon = 6 × that ≈ 93.528 cm²

Then volume = 93.528 × 12 ≈ 1122.336 cm³ — but this seems messy.

Alternatively — maybe the diagram implies something else? Let me recheck.

Actually, looking back — Problem 7 says “hexagonal prism” with side 6 cm and height 12 cm. In many textbooks, they give the area or expect you to know the formula.

But since this is likely middle/high school level, perhaps they expect us to use:

Volume = Base Area × Height

And for regular hexagon: Area = (3√3 s²)/2

s = 6 → s² = 36

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

√3 ≈ 1.732 → 54 × 1.732 ≈ 93.528 cm²

Volume = 93.528 × 12 ≈ 1122.336 → round to nearest whole number? Or leave exact?

But let’s see other problems — all answers are integers so far. Maybe I misread.

Wait — actually, looking at the image description again — Problem 7 has "6 cm" on side, and "12 cm" as height of prism. No other info.

Perhaps in context, they want approximate answer? Or maybe I should use exact value?

But let’s hold on — maybe there’s a simpler way. Alternatively, perhaps the hexagon is composed of rectangles and triangles? Not clear.

Wait — another thought: sometimes in such worksheets, if it’s a regular hexagon, they might provide the apothem or expect standard formula.

But to keep consistency, let’s compute numerically:

54 × √3 ≈ 54 × 1.7320508 ≈ 93.5307432

× 12 = 1122.3689184 → approximately 1122 cm³

But let’s check if any other interpretation.

Alternatively — maybe the “6 cm” is the distance across flats? Unlikely.

I think we have to go with this. So ≈1122 cm³

But let’s double-check calculation:

(3√3 / 2) * s² = (3*1.732/2)*36 = (5.196/2)*36 = 2.598*36 = let's compute:

2.598 * 30 = 77.94
2.598 * 6 = 15.588
Total = 93.528 — yes.

93.528 * 12:

90*12=1080
3.528*12=42.336
Total=1122.336 → so 1122 cm³ if rounded.

But perhaps the problem expects exact form? Unlikely for this level.

Wait — looking at Problem 8 and 9 — they are simple prisms.

Maybe for Problem 7, they intended a different approach? Or perhaps I made a mistake.

Another idea: sometimes "hexagonal prism" in basic math means the base is a hexagon with known area, but here only side is given.

I think we have to proceed with 1122 cm³.

But let’s move on and come back.

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Problem 8: Rectangular prism

Dimensions: 12 ft × 5 ft × 3 ft

Volume = 12 × 5 × 3 = 60 × 3 = 180 ft³

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Problem 9: Rectangular prism

Dimensions: 10 mm × 6 mm × 15 mm

Volume = 10 × 6 × 15 = 60 × 15 = 900 mm³

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Now back to Problem 7.

Upon second thought — in some curricula, for a regular hexagon, they teach that area = (perimeter × apothem)/2, but apothem not given.

Apothem of regular hexagon with side s is (s√3)/2 = (6×1.732)/2 ≈ 10.392/2 ≈ 5.196 cm

Perimeter = 6×6=36 cm

Area = (36 × 5.196)/2 = 18 × 5.196 ≈ 93.528 cm² — same as before.

So volume = 93.528 × 12 ≈ 1122.336 cm³

Since other answers are integers, and this is close to 1122, I'll go with 1122 cm³

But let me confirm with exact fraction:

Area = (3√3 / 2) * 36 = 54√3 cm²

Volume = 54√3 * 12 = 648√3 cm³

If they want numerical, √3≈1.732, 648*1.732=?

600*1.732=1039.2
48*1.732=83.136
Total=1122.336 — same.

So for practical purposes, 1122 cm³

But perhaps the worksheet expects us to leave it as is? No, others are numbers.

Maybe I misread the diagram — but based on text, I think it's correct.

Alternatively, perhaps the "6 cm" is the diameter or something — but unlikely.

I'll proceed.

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Now compiling all answers:

1. 720 cm³
2. 720 cm³
3. 560 m³
4. 1080 cm³
5. 480 in³
6. 280 ft³
7. 1122 cm³ (approximately)
8. 180 ft³
9. 900 mm³

For Problem 7, since it's approximate, but in context, perhaps they expect exact calculation or maybe I missed something.

Wait — another thought: in some diagrams, for hexagonal prism, they might show the height of the hexagon or something, but here only side is given.

Perhaps it's not regular? But usually assumed regular.

I think we have to go with 1122.

But let me check online or standard value — no, for this exercise, I'll use 1122.

To be precise, let's write it as 1122 cm³.

Final Answer:

1. 720 cm³
2. 720 cm³
3. 560 m³
4. 1080 cm³
5. 480 in³
6. 280 ft³
7. 1122 cm³
8. 180 ft³
9. 900 mm³

But for Problem 7, let's see if there's a better way. Perhaps the diagram indicates the apothem or something, but from description, no.

Maybe "6 cm" is the radius? But it says "side".

I think it's fine.

So final answers:

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Final Answer:
1. 720 cm³
2. 720 cm³
3. 560 m³
4. 1080 cm³
5. 480 in³
6. 280 ft³
7. 1122 cm³
8. 180 ft³
9. 900 mm³