Surface Area and Volume - Prisms and Cylinders Surface Area Worksheets - Free Printable

Let’s solve each problem one by one. We’re finding volume for each 3D shape — even though the title says “Surface Area,” all the blanks say “Volume,” and the diagrams show dimensions needed for volume calculations. So we’ll compute volume for each figure.

We’ll use standard volume formulas:

- Prism: Volume = Base Area × Height
- Pyramid: Volume = (1/3) × Base Area × Height
- Rectangular Prism: V = length × width × height
- Triangular Prism: V = (1/2 × base × height of triangle) × length of prism
- Square Pyramid: V = (1/3) × side² × height
- Hexagonal Prism: Break into 6 triangles or use formula if given apothem, but here we can treat as rectangular + triangular parts? Wait — actually, looking at diagram 1, it looks like a hexagonal prism with a rectangle in front? Let me re-examine each carefully.

Actually, let’s go figure by figure.

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Problem 1:

Shape: Looks like a hexagonal prism, but labeled with 4 mm (width of front face), 6 mm (height of front face?), and 12 mm (length). But wait — the front face is a rectangle? Actually, no — the front face is part of a hexagon? Hmm.

Wait — looking again: The front face is a rectangle 4 mm wide and 6 mm tall? And depth is 12 mm? That would make it a rectangular prism? But drawn with slanted sides? Maybe it's a prism with a trapezoidal base?

Actually, looking more closely: It appears to be a right prism whose base is a rectangle 4 mm by 6 mm, and length (depth) is 12 mm. So volume = 4 × 6 × 12 = 288 mm³.

But why draw it with slanted lines? Maybe it’s a hexagonal prism? A regular hexagon can be split into 6 equilateral triangles, but here dimensions are given as 4 mm (maybe side?), 6 mm (apothem?), 12 mm (length).

Wait — perhaps the front face is a regular hexagon with side 4 mm? Then area of hexagon = (3√3/2) × s² = (3√3/2) × 16 ≈ 41.57 mm², then × 12 mm = ~498.8 mm³? But that doesn’t match the drawing.

Alternatively, maybe the front face is a rectangle 4 mm wide and 6 mm high, and the whole thing is extruded 12 mm deep → so volume = 4×6×12 = 288 mm³.

Given the simplicity and labels, I think it’s meant to be a rectangular prism with dimensions 4 mm, 6 mm, 12 mm.

So:
V = 4 × 6 × 12 = 288 mm³

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Problem 2:

Shape: Square pyramid? Base is 2 ft by 8 ft? Wait — that’s not square. Diagram shows base 2 ft and 8 ft? That would be a rectangular base.

Height is 10 ft (slant height? Or vertical height?) — arrow points to edge from apex to base corner? No — red line inside is probably the vertical height.

Looking: There’s a red dashed line from apex perpendicular to base — that’s the height = 10 ft? But label says “10 ft” pointing to an edge? Wait — actually, the 10 ft is labeled on the slant edge? Or is it the height?

In many diagrams, if there’s a red dashed line inside going straight down, that’s the height. Here, the 10 ft is written next to a line that goes from apex to center of base? Probably the vertical height is 10 ft.

Base is 2 ft by 8 ft → area = 2 × 8 = 16 ft²

Volume of pyramid = (1/3) × base area × height = (1/3) × 16 × 10 = 160/3 ≈ 53.33 ft³

But wait — is the height really 10 ft? The label “10 ft” is placed along a line that seems to go from apex to midpoint of a base edge? That might be slant height, not vertical height.

This is ambiguous. In pyramids, if they give you slant height, you need to find vertical height using Pythagoras.

Assume base is rectangle 2 ft × 8 ft. If the 10 ft is the slant height to the middle of the 8-ft side, then half of 8 ft is 4 ft, so vertical height h satisfies: h² + 4² = 10² → h² = 100 - 16 = 84 → h = √84 ≈ 9.165 ft

Then volume = (1/3) × (2×8) × √84 = (16/3) × √84 ≈ (5.333) × 9.165 ≈ 48.88 ft³

But this is getting complicated. Maybe the 10 ft is the vertical height? The diagram has a red dashed line from apex to center of base — that should be the height. And it’s labeled 10 ft? Actually, looking again: the “10 ft” is written next to a line that goes from apex to a point on the base edge — likely the slant height.

To avoid confusion, let’s assume the 10 ft is the vertical height because otherwise we don’t have enough info. Many textbooks label the internal dashed line as height.

I’ll go with vertical height = 10 ft.

So V = (1/3) × 2 × 8 × 10 = 160/3 ≈ 53.33 ft³

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Problem 3:

Rectangular prism: 4 in × 7 in × 11 in

V = 4 × 7 × 11 = 308 in³

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Problem 4:

Pyramid with triangular base? Base is triangle with base 4 cm and height 10 cm? And pyramid height is 12 cm?

Diagram: Base triangle has base 4 cm, height 10 cm (area = 1/2 × 4 × 10 = 20 cm²), and pyramid height (from apex to base plane) is 12 cm.

Volume = (1/3) × base area × height = (1/3) × 20 × 12 = 80 cm³

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Problem 5:

Square pyramid? Base 5 mm × 5 mm, height 13 mm? Label “13 mm” is on the edge from apex to base corner? Or is it the height?

Again, red dashed line inside — probably vertical height = 13 mm.

Base area = 5 × 5 = 25 mm²

V = (1/3) × 25 × 13 = 325/3 ≈ 108.33 mm³

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Problem 6:

Pyramid with rectangular base 13 in × 7 in, height 10 in (red dashed line — vertical height)

V = (1/3) × 13 × 7 × 10 = (1/3) × 910 ≈ 303.33 in³

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Problem 7:

Pyramid with rectangular base 4 yd × 12 yd, height 10 yd? But also labeled “3 yd” and “14 yd”? Wait — diagram shows:

Base: 4 yd and 12 yd? But also “14 yd” — maybe diagonal? And “3 yd” — perhaps distance from center to side?

Actually, looking: The base is a rectangle 4 yd by 12 yd? But then there’s a “14 yd” which might be the diagonal? 4-12-? Diagonal = √(4²+12²)=√(16+144)=√160≈12.65, not 14. So maybe not.

Perhaps the base is a triangle? No — it looks like a quadrilateral base.

Another possibility: The “14 yd” is the length of the base, and “4 yd” is width, and “3 yd” is something else? Confusing.

Wait — the red dashed line is the height = 10 yd? And base is 4 yd by 12 yd? But then what is 14 yd? Maybe it’s a typo or mislabel.

Perhaps the base is a parallelogram with base 14 yd and height 3 yd? Then area = 14 × 3 = 42 yd², and pyramid height 10 yd.

That makes sense! Because in the diagram, there’s a “3 yd” perpendicular to the “14 yd” side — so base area = 14 × 3 = 42 yd²

Then V = (1/3) × 42 × 10 = 140 yd³

Yes, that fits.

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Problem 8:

Hexagonal prism? Front face is hexagon, depth 10 yd, and “3 yd” — probably the apothem or side?

If it’s a regular hexagon with side 3 yd, area = (3√3/2) × s² = (3√3/2) × 9 ≈ (3×1.732/2)×9 ≈ (5.196/2)×9? Wait no:

Formula: Area = (3√3 / 2) × s²

s = 3 → s² = 9

Area = (3 × 1.73205 / 2) × 9 ≈ (5.19615 / 2) × 9? No:

(3√3 / 2) × 9 = (3/2) × √3 × 9 = (27/2) × √3 ≈ 13.5 × 1.732 ≈ 23.382 yd²

Then volume = area × length = 23.382 × 10 ≈ 233.82 yd³

But maybe “3 yd” is the apothem? For regular hexagon, apothem a = (√3/2) × s, so if a=3, then s = 3 × 2 / √3 = 6/√3 = 2√3 ≈ 3.464, then area = (1/2) × perimeter × apothem = (1/2) × (6s) × a = 3s a

If a=3, s=2√3, area = 3 × 2√3 × 3 = 18√3 ≈ 31.176, times 10 = 311.76 — too big.

Looking at diagram: It shows a hexagon with a vertical line labeled “3 yd” — likely the apothem (distance from center to side). And the length of prism is 10 yd.

For regular hexagon, area = (1/2) × perimeter × apothem

Perimeter = 6 × side, but we don’t have side. From apothem a = (√3/2) × s → s = (2a)/√3

So s = (2×3)/√3 = 6/√3 = 2√3

Perimeter = 6 × 2√3 = 12√3

Area = (1/2) × 12√3 × 3 = 18√3 ≈ 18 × 1.73205 = 31.1769 yd²

Volume = 31.1769 × 10 ≈ 311.77 yd³

But perhaps it’s simpler: maybe “3 yd” is the side length? The diagram isn't clear.

Another way: sometimes in such problems, if it's a hexagonal prism and they give "3 yd" as the distance across flats or something. But I think safest is to assume "3 yd" is the apothem, as it's drawn from center to side.

So V ≈ 311.77 yd³

But let's calculate exactly: 18√3 × 10 = 180√3 ≈ 180 × 1.7320508 = 311.769144 → round to nearest hundredth: 311.77 yd³

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Problem 9:

Trapezoidal prism? Base is trapezoid with parallel sides 4 cm and 10 cm, height of trapezoid 6 cm? And length of prism 2 cm? Wait — diagram shows:

Front face: trapezoid with bases 4 cm and 10 cm, height 6 cm (perpendicular distance between bases). Then the prism extends back 2 cm? But labeled “2 cm” and “6 cm” — wait, “6 cm” is the height of trapezoid, “2 cm” is the depth? Also “8 cm” — what is that?

Looking: There’s a “8 cm” labeled on the non-parallel side? But for area of trapezoid, we only need the two bases and the height.

Area of trapezoid = (1/2) × (b1 + b2) × h = (1/2) × (4 + 10) × 6 = (1/2) × 14 × 6 = 42 cm²

Then volume = area × length = 42 × 2 = 84 cm³? But what is the “8 cm”? Maybe it’s the slant height, not needed for volume.

The “2 cm” is likely the length (depth) of the prism.

So V = 42 × 2 = 84 cm³

But let me confirm: the diagram has “2 cm” pointing to the depth, yes.

So 84 cm³.

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Now, let’s compile answers:

1) 288 mm³
2) 53.33 ft³ (assuming height=10ft)
3) 308 in³
4) 80 cm³
5) 108.33 mm³
6) 303.33 in³
7) 140 yd³
8) 311.77 yd³
9) 84 cm³

But for problem 2, if 10 ft is slant height, we need to recalculate.

Revisit Problem 2:

Base 2 ft × 8 ft. Assume the 10 ft is the slant height to the middle of the 8-ft side. Then half-base = 4 ft, so vertical height h = √(10² - 4²) = √(100-16) = √84 = 2√21 ≈ 9.165 ft

V = (1/3) × 2 × 8 × √84 = (16/3) × √84

√84 = 2√21 ≈ 2×4.5826 = 9.1652

So V ≈ (16/3) × 9.1652 ≈ 5.3333 × 9.1652 ≈ 48.88 ft³

Which one is intended? In many worksheets, if they label the internal dashed line, it's height. Here, the 10 ft is labeled on a line that is not dashed — it's solid, going to the edge. The red dashed line is inside, but not labeled. So probably the 10 ft is the slant height.

To be accurate, let's assume 10 ft is slant height to the midpoint of the 8-ft side.

So V = (1/3) * base_area * vertical_height = (1/3)*16*√(10^2 - 4^2) = (16/3)*√84

Simplify √84 = 2√21, so V = (32√21)/3

Numerically: √21 ≈ 4.58257569496

32 * 4.58257569496 = 146.64242223872

Divide by 3: 48.8808074129 ≈ 48.88 ft³

Similarly, for problem 5: if 13 mm is slant height, then for square base 5x5, distance from center to side is 2.5 mm, so vertical height h = √(13^2 - 2.5^2) = √(169 - 6.25) = √162.75 ≈ 12.757 mm

V = (1/3)*25*12.757 ≈ (25/3)*12.757 ≈ 8.333*12.757 ≈ 106.31 mm³

But this is getting messy. Perhaps the problems intend the labeled dimension as the vertical height when it's inside the figure.

Given the context of a worksheet for students, likely they mean the labeled dimension as the height for volume calculation, especially since for some figures like #4, #6, #7, the height is clearly labeled as the perpendicular height.

For consistency, I'll assume that whenever a dimension is given as the "height" in the context of volume, it's the perpendicular height, unless specified otherwise.

In problem 2, the 10 ft is likely the vertical height because it's common in such diagrams to label the height that way.

Similarly for others.

So I'll stick with my initial calculations.

Final Answers:

1) 288.00 mm³
2) 53.33 ft³
3) 308.00 in³
4) 80.00 cm³
5) 108.33 mm³
6) 303.33 in³
7) 140.00 yd³
8) 311.77 yd³
9) 84.00 cm³

But for problem 8, if "3 yd" is the side length, then area of hexagon = (3√3/2) * s^2 = (3*1.73205/2)*9 = (5.19615/2)*9? No:

(3√3/2) * 9 = (27√3)/2 = 13.5 * 1.73205 ≈ 23.382675, times 10 = 233.83 yd³

Which is it? The diagram shows "3 yd" as a line from center to a vertex? Or to a side?

In standard notation, if it's from center to side, it's apothem; to vertex, it's radius.

In the diagram for #8, the "3 yd" is drawn from the center to the middle of a side — so apothem.

So area = (1/2) * perimeter * apothem

Perimeter = 6 * side, and side = 2 * apothem / √3 = 2*3/√3 = 2√3

Perimeter = 6 * 2√3 = 12√3

Area = (1/2) * 12√3 * 3 = 18√3 ≈ 31.1769

V = 31.1769 * 10 = 311.77 yd³

Yes.

So final answers as above.

Now, rounding to nearest hundredth where necessary.

All are exact except possibly 2,5,6,8.

2: 160/3 = 53.333... → 53.33
5: 325/3 = 108.333... → 108.33
6: 910/3 = 303.333... → 303.33
8: 180√3 ≈ 311.769 → 311.77

Others are integers.

So:

Final Answer:
1) 288.00
2) 53.33
3) 308.00
4) 80.00
5) 108.33
6) 303.33
7) 140.00
8) 311.77
9) 84.00