Surface Area Practice: Prisms and Pyramids | Worksheet - Free Printable

Let’s solve each problem step by step. We’ll find the surface area of each prism or pyramid using the right formulas.

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Problem 1: Rectangular Prism (8 ft × 4 ft × 4 ft)
Surface Area = 2(lw + lh + wh)
= 2(8×4 + 8×4 + 4×4)
= 2(32 + 32 + 16)
= 2(80) = 160 ft²

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Problem 2: Triangular Prism (base triangle: 3 in, 4 in, 5 in; height of prism = 8 in)
First, check if it’s a right triangle: 3-4-5 → yes! So area of base = (3×4)/2 = 6 in²
Lateral faces: three rectangles
- 3 in × 8 in = 24 in²
- 4 in × 8 in = 32 in²
- 5 in × 8 in = 40 in²
Total lateral area = 24 + 32 + 40 = 96 in²
Two bases: 2 × 6 = 12 in²
Total Surface Area = 96 + 12 = 108 in²

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Problem 3: Cube (10 mm on each side)
Surface Area = 6 × (side)² = 6 × 100 = 600 mm²

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Problem 4: Rectangular Prism (9 in × 2 in × 6 in)
SA = 2(lw + lh + wh)
= 2(9×2 + 9×6 + 2×6)
= 2(18 + 54 + 12)
= 2(84) = 168 in²

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Problem 5: Square Pyramid (base 4 ft × 4 ft, slant height 9 ft)
Base area = 4 × 4 = 16 ft²
Each triangular face: (base × slant height)/2 = (4 × 9)/2 = 18 ft²
Four faces: 4 × 18 = 72 ft²
Total SA = 16 + 72 = 88 ft²

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Problem 6: Square Pyramid (base 3 in × 3 in, slant height 8 in)
Base area = 3 × 3 = 9 in²
Each triangular face: (3 × 8)/2 = 12 in²
Four faces: 4 × 12 = 48 in²
Total SA = 9 + 48 = 57 in²

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Problem 7: Square Pyramid (base 10 mm × 10 mm, slant height 12 mm)
Wait — diagram shows two different slant heights? Actually, looking closely:
It says “12 mm” for the slant edge from apex to base corner? But that’s not slant height.
Actually, in pyramids, we need the *slant height* (height of triangular face), not the edge length.
But here, it labels “12 mm” as the edge from apex to base vertex, and “8.7 mm” as half-diagonal? That doesn’t help directly.

Wait — let me re-read: The figure is labeled with:
- Base: 10 mm × 10 mm
- Two edges from apex to base corners: 12 mm each
- One label “8.7 mm” — likely the distance from center of base to midpoint of a side? Or maybe it’s the apothem?

Actually, this might be a trick. If it’s a square pyramid and they give you the edge from apex to base corner (12 mm) and base side 10 mm, then we can compute the actual slant height.

Let’s do that:

In a square pyramid, the slant height (l) is the height of the triangular face. To find it, consider the right triangle formed by:
- Height of pyramid (h)
- Distance from center of base to middle of a side = half the base = 5 mm
- Slant height (l) — hypotenuse of that triangle

But we don’t have h. Instead, we have the edge from apex to base corner = 12 mm.

That edge goes from apex to a corner of the base. The distance from center of base to a corner is half the diagonal of the base.

Diagonal of base = √(10² + 10²) = √200 ≈ 14.142 mm → half = 7.071 mm

Now, in the right triangle: apex to center (height h), center to corner (7.071 mm), and edge (12 mm) as hypotenuse.

So: h² + (7.071)² = 12²
h² + 50 = 144 → h² = 94 → h ≈ 9.7 mm

But we need slant height — which is from apex to midpoint of a base side.

Distance from center to midpoint of side = 5 mm (half of 10 mm)

Then slant height l = √(h² + 5²) = √(94 + 25) = √119 ≈ 10.908 mm

But wait — the diagram also says “8.7 mm”. Maybe that’s the slant height? Let me check: 8.7² = 75.69, and 5² = 25, so h² = 75.69 - 25 = 50.69 → h ≈ 7.12, then edge to corner would be √(50.69 + 50) = √100.69 ≈ 10.03 — not 12. Doesn’t match.

Alternatively, perhaps “8.7 mm” is meant to be the slant height? But 8.7 seems too small compared to 12.

Looking again at the diagram description: It says “12 mm” twice (edges to corners), and “8.7 mm” near the base — possibly the apothem of the base? No.

Wait — maybe it's a typo or mislabel. In many textbooks, for a square pyramid with base 10 and lateral edge 12, they expect you to calculate slant height properly.

But let’s try another way: Perhaps “8.7 mm” is the slant height? Let’s test: if slant height = 8.7, then area of one triangle = (10 × 8.7)/2 = 43.5, four triangles = 174, base = 100, total = 274. But that ignores the 12 mm label.

Alternatively, maybe the 12 mm is the slant height? Then why label 8.7?

I think there’s confusion in the diagram labeling. Let me look at Problem 8 for comparison.

Problem 8: Square pyramid, base 9 in × 9 in, slant height 9 in — straightforward.

Back to Problem 7: Given the ambiguity, but since it’s a practice sheet, likely they intend “12 mm” as the slant height? But it’s drawn as the edge.

Wait — in some diagrams, they label the slant height along the face. Let me assume that “12 mm” is the slant height (even though it’s drawn to the corner). Or perhaps “8.7 mm” is irrelevant?

Another idea: 8.7 mm might be the height of the pyramid? Let’s try that.

If height h = 8.7 mm, base 10x10, then distance from center to side = 5 mm, so slant height l = √(8.7² + 5²) = √(75.69 + 25) = √100.69 ≈ 10.03 mm

Then area of one triangle = (10 × 10.03)/2 ≈ 50.15, four = 200.6, base = 100, total ≈ 300.6 — messy.

Perhaps the “12 mm” is the slant height, and “8.7 mm” is a distractor or misprint.

Given that in Problem 5 and 6, they clearly label slant height, and here they have “12 mm” on the face, I’ll assume 12 mm is the slant height.

So for Problem 7:
Base = 10 mm × 10 mm = 100 mm²
Each triangular face = (10 × 12)/2 = 60 mm²
Four faces = 240 mm²
Total SA = 100 + 240 = 340 mm²

And ignore the 8.7 mm as possibly a mistake or for another purpose.

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Problem 8: Square Pyramid (base 9 in × 9 in, slant height 9 in)
Base area = 9 × 9 = 81 in²
Each triangular face = (9 × 9)/2 = 40.5 in²
Four faces = 4 × 40.5 = 162 in²
Total SA = 81 + 162 = 243 in²

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Final Answers:

1. 160 ft²
2. 108 in²
3. 600 mm²
4. 168 in²
5. 88 ft²
6. 57 in²
7. 340 mm² (assuming 12 mm is slant height)
8. 243 in²

Note: For Problem 7, if the 12 mm is actually the edge to the corner and not slant height, the answer would be different, but based on typical textbook problems and the other questions, 12 mm is likely intended as slant height.

Final Answer:
1. 160 ft²
2. 108 in²
3. 600 mm²
4. 168 in²
5. 88 ft²
6. 57 in²
7. 340 mm²
8. 243 in²