Surface Area of a Prism - Go Teach Maths: Handcrafted Resources - Free Printable

Let’s solve each problem one by one. We’ll use the formula for volume:

- For prisms: Volume = Base Area × Height
- For cylinders: Volume = π × r² × h (where r is radius, h is height)

We’ll round to the nearest hundredth if needed.

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Problem 1)
This is a triangular prism. The base is a right triangle with legs 5 ft and 5 ft.
Area of triangle = (1/2) × base × height = (1/2) × 5 × 5 = 12.5 ft²
Height of prism = 10 ft
Volume = 12.5 × 10 = 125.00 ft³

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Problem 2)
Cylinder. Radius = 8 ft, Height = 12 ft
Volume = π × 8² × 12 = π × 64 × 12 = 768π ≈ 768 × 3.1416 ≈ 2412.74 ft³

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Problem 3)
Cube! All sides are 6 in.
Volume = side³ = 6 × 6 × 6 = 216.00 in³

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Problem 4)
Rectangular prism. Dimensions: 10 cm × 8 cm × 3 cm
Volume = length × width × height = 10 × 8 × 3 = 240.00 cm³

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Problem 5)
Trapezoidal prism. First find area of trapezoid base.
Bases: 6 yd and 13 yd, height of trapezoid = 7 yd
Area = (1/2) × (6 + 13) × 7 = (1/2) × 19 × 7 = 66.5 yd²
Length of prism (height) = 8 yd
Volume = 66.5 × 8 = 532.00 yd³

Wait — let me double-check the diagram. The “8 yd” is labeled as the length of the prism, yes. And the trapezoid has bases 6 and 13, height 7. Correct. So 66.5 × 8 = 532. Yes.

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Problem 6)
Hexagonal prism? Wait — it looks like a hexagon base, but they give us apothem or something? Actually, looking at labels:
It says “3 in” (probably height of prism), “2 in” (apothem?), “10 in” (side length?). But wait — actually, this might be a regular hexagon. But we don’t have enough info unless... Hmm.

Actually, re-examining: It shows a hexagonal prism. The base is a regular hexagon. They label “10 in” as the distance across flats? Or side? Wait — there’s a “2 in” pointing to the apothem (distance from center to middle of side), and “10 in” is probably the side length? No — that doesn’t make sense.

Wait — look again: There’s a line labeled “2 in” going from center to midpoint of a side → that’s the apothem. And “10 in” is the full width across two opposite sides? That would mean the diameter across flats is 10 in, so apothem should be half of that? But they say apothem is 2 in? Contradiction.

Wait — maybe I misread. Let me interpret differently.

Actually, in many worksheets, when they show a hexagon with “2 in” as the apothem and “10 in” as the perimeter? No.

Wait — perhaps “10 in” is the side length? But then apothem for regular hexagon = (√3/2) × side ≈ 0.866 × 10 = 8.66, not 2. Doesn’t match.

Alternative interpretation: Maybe “10 in” is the distance across corners (diameter), and “2 in” is apothem? Still inconsistent.

Wait — perhaps it's NOT a regular hexagon? Or maybe the “2 in” is the height of a triangle inside?

Looking more carefully: The figure shows a hexagonal prism. Inside the base, they drew lines from center to vertices, making 6 triangles. One triangle has height “2 in” (from center to side) and base “10 in”? That can't be — because if base of triangle is 10 in, and height 2 in, area of one triangle = (1/2)*10*2 = 10 in², times 6 = 60 in² for base. Then volume = 60 × 3 = 180 in³.

But is the “10 in” the side of the hexagon or the base of the triangle? In a regular hexagon, if you divide into 6 equilateral triangles, the side of the hexagon equals the radius. But here, if they’re showing a triangle with base 10 in and height 2 in, that suggests the side of the hexagon is 10 in? But then height shouldn’t be 2 in.

I think there’s a mistake in my assumption. Let me check standard approach.

Actually, in some diagrams, they label the “apothem” as the perpendicular distance from center to a side, and the “radius” as to vertex. But here, they have “2 in” as apothem, and “10 in” might be the side length? But for regular hexagon, apothem = (√3/2) * side. If side = s, apothem = (√3/2)s. Set equal to 2: s = 4/√3 ≈ 2.309, not 10.

Alternatively, perhaps “10 in” is the perimeter? 10/6 ≈ 1.666 per side, apothem 2 — still doesn’t fit.

Wait — another idea: Maybe the “10 in” is the length of the prism? No, it’s labeled on the base.

Looking back at the image description: “6) [hexagonal prism] 3 in (height), 2 in (apothem?), 10 in (side?)”

Perhaps it’s a typo or mislabel. But in many school problems, they simplify: if they give apothem and perimeter, area = (1/2) × apothem × perimeter.

Do they give perimeter? Not directly. But if “10 in” is the side length, and it’s regular hexagon, perimeter = 6 × 10 = 60 in. Apothem = 2 in. Then area = (1/2) × 2 × 60 = 60 in². Then volume = 60 × 3 = 180 in³.

That makes sense numerically, even if geometrically inconsistent (because for side 10, apothem should be ~8.66, not 2). But since this is a worksheet, likely they intend: apothem = 2 in, and “10 in” is meant to be the side length, so perimeter = 60 in.

So I’ll go with that.

Base area = (1/2) × apothem × perimeter = (1/2) × 2 × (6×10) = (1/2)×2×60 = 60 in²
Height of prism = 3 in
Volume = 60 × 3 = 180.00 in³

*(Note: This assumes "10 in" is side length of hexagon, even though apothem doesn't match real geometry — common simplification in worksheets.)*

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Problem 7)
Hexagonal prism again. Labels: “7 mm” (probably apothem), “13 mm” (side length?).
Assume regular hexagon. Side = 13 mm, apothem = 7 mm.
Perimeter = 6 × 13 = 78 mm
Base area = (1/2) × apothem × perimeter = (1/2) × 7 × 78 = 7 × 39 = 273 mm²
Height of prism? Not given? Wait — the figure shows only base dimensions. Oh no — missing height?

Wait — looking back: In problem 7, it’s a hexagonal prism, labeled “7 mm” and “13 mm”, but no height? That can’t be.

Re-examining user input: “7) [hexagonal prism] 7 mm, 13 mm” — probably 7 mm is height of prism, 13 mm is side? Or vice versa?

In the diagram description, it might be that “7 mm” is the height of the prism, and “13 mm” is the side of the hexagon. But we need apothem to find area.

For regular hexagon with side s, apothem = (√3/2) × s ≈ 0.866 × s.
If s = 13 mm, apothem ≈ 0.866 × 13 ≈ 11.258 mm. But they didn’t give that.

Perhaps “7 mm” is the apothem? Then if apothem = 7, and side = ? From apothem = (√3/2)s ⇒ s = 2×apothem / √3 = 14 / 1.732 ≈ 8.08 mm. But they gave 13 mm — conflict.

Another possibility: “13 mm” is the distance across flats (i.e., twice the apothem)? So apothem = 13/2 = 6.5 mm? But they labeled “7 mm” separately.

I think there’s confusion. Let me assume based on common problems: Often, for hexagonal prism, they give side length and height. Here, “13 mm” is likely the side length, and “7 mm” is the height of the prism. But we need apothem to find base area.

Since it’s regular hexagon, area = (3√3/2) × s²
s = 13 mm
Area = (3 × 1.732 / 2) × 169 ≈ (5.196 / 2) × 169 ≈ 2.598 × 169 ≈ 439.062 mm²
Then volume = area × height = 439.062 × 7 ≈ 3073.43 mm³

But they didn’t specify which is which. Alternatively, if “7 mm” is apothem, and “13 mm” is side, but as before, inconsistent.

Wait — in the original problem list, problem 6 had similar issue. Perhaps for consistency, in problem 7, “7 mm” is apothem, “13 mm” is side, and we use area = (1/2)*apothem*perimeter.

Perimeter = 6 × 13 = 78 mm
Apothem = 7 mm
Base area = (1/2) × 7 × 78 = 273 mm²
Now, what is the height of the prism? Not labeled! Oh no.

Looking back at user input: “7) [figure] 7 mm, 13 mm” — probably both are for the base, and height is missing? That can’t be.

Perhaps “7 mm” is the height of the prism, and “13 mm” is the side of the hexagon. Then we need to calculate base area using formula for regular hexagon.

Standard formula: Area = (3√3/2) * s^2
s = 13 mm
Area = (3 * √3 / 2) * 169 = (3 * 1.73205 / 2) * 169 ≈ (5.19615 / 2) * 169 = 2.598075 * 169 ≈ 439.074675 mm²
Height = 7 mm (assuming)
Volume = 439.074675 * 7 ≈ 3073.52 mm³

But to be precise, let's use exact values.

√3 ≈ 1.7320508
3√3/2 = (3*1.7320508)/2 = 5.1961524/2 = 2.5980762
Times 169 = 2.5980762 * 169 = let's compute:
2.5980762 * 170 = 441.672954, minus 2.5980762 = 439.0748778
Times 7 = 3073.5241446 ≈ 3073.52 mm³

But I'm not sure if "7 mm" is height. Perhaps in the diagram, "7 mm" is labeled as the height of the prism, and "13 mm" as side of base. I'll go with that.

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Problem 8)
Cylinder. Radius = 6 cm, Height = 8 cm
Volume = π × r² × h = π × 36 × 8 = 288π ≈ 288 × 3.1416 ≈ 904.78 cm³

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Problem 9)
Triangular prism. Base is triangle with base 3 yd, height 2 yd? Wait — labels: “2 yd” (height of triangle?), “3 yd” (base of triangle?), “5 yd” (hypotenuse?), “10 yd” (length of prism).

The triangle has sides 2, 3, 5? But 2+3=5, which is degenerate — not possible. Must be right triangle? 2 and 3 are legs, 5 is hypotenuse? But 2² + 3² = 4+9=13 ≠ 25. Not right triangle.

Perhaps “2 yd” is height of triangle, “3 yd” is base, and “5 yd” is slant or something else. But for area of triangle, we need base and height.

Likely, the triangle has base 3 yd and height 2 yd (since they are perpendicular, as shown by right angle symbol?).

In the description: “9) [triangular prism] 2 yd, 3 yd, 5 yd, 10 yd” — probably 2 yd and 3 yd are legs of right triangle, 5 yd is hypotenuse (but 2-3-5 not Pythagorean), or perhaps 3 yd is base, 2 yd is height, 5 yd is not needed, 10 yd is length of prism.

Assume the triangular base has base = 3 yd, height = 2 yd (area = (1/2)*3*2 = 3 yd²), and length of prism = 10 yd.

Volume = 3 × 10 = 30.00 yd³

The “5 yd” might be a red herring or mislabeled, but since 2 and 3 are given with right angle, likely those are the legs.

Confirm: If it's a right triangle with legs 2 and 3, area = 3, times length 10 = 30. Yes.

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Now, compiling all answers:

1) 125.00 ft³
2) 2412.74 ft³
3) 216.00 in³
4) 240.00 cm³
5) 532.00 yd³
6) 180.00 in³ (with assumption)
7) 3073.52 mm³ (assuming 13 mm side, 7 mm height)
8) 904.78 cm³
9) 30.00 yd³

But for problem 7, I'm uncertain. Let me double-check common practices.

Upon second thought, in many textbooks, for hexagonal prism, if they give "side" and "height", they expect you to use the formula. But here, since problem 6 used apothem and side, perhaps for 7, "7 mm" is apothem, "13 mm" is side, and height is not given? That can't be.

Looking back at the user's initial request, the image is described, but in text, for problem 7: "7 mm" and "13 mm" — likely "7 mm" is the height of the prism, and "13 mm" is the side of the hexagon. And we must calculate base area.

To avoid error, let's use the standard formula for regular hexagon area: A = (3√3/2) * s^2

s = 13 mm
A = (3 * √3 / 2) * 169
√3 ≈ 1.73205080757
3 * 1.73205080757 = 5.19615242271
Divide by 2: 2.59807621135
Times 169: 2.59807621135 * 169 = let's calculate:

2.59807621135 * 100 = 259.807621135
2.59807621135 * 60 = 155.884572681
2.59807621135 * 9 = 23.38268590215
Sum: 259.807621135 + 155.884572681 = 415.692193816
+23.38268590215 = 439.07487971815 mm²

Height = 7 mm (assumed)
Volume = 439.07487971815 * 7 = 3073.52415802705 ≈ 3073.52 mm³

Yes.

For problem 6, similarly, if "10 in" is side, "2 in" is apothem, but geometrically impossible, but for worksheet, we use area = (1/2)*apothem*perimeter = (1/2)*2*(6*10) = 60 in², times height 3 in = 180 in³.

I think that's intended.

Final Answers:

1) 125.00
2) 2412.74
3) 216.00
4) 240.00
5) 532.00
6) 180.00
7) 3073.52
8) 904.78
9) 30.00

With units as given.

Now, writing final answer as requested.

Final Answer:
1) 125.00 ft³
2) 2412.74 ft³
3) 216.00 in³
4) 240.00 cm³
5) 532.00 yd³
6) 180.00 in³
7) 3073.52 mm³
8) 904.78 cm³
9) 30.00 yd³