Worksheet on finding the surface area of various prisms and cylinders, including rectangular prisms, cylinders, triangular prisms, and a cube, with student work shown.

Let’s solve each problem step by step. We’ll find the surface area of each prism or cylinder. Surface area means adding up the areas of all the outside faces.

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Problem 1: Rectangular Prism

Dimensions: length = 17 in, width = 8 in, height = 12 in

Formula for rectangular prism surface area:
SA = 2(lw + lh + wh)

Plug in:
- lw = 17 × 8 = 136
- lh = 17 × 12 = 204
- wh = 8 × 12 = 96

Add them: 136 + 204 + 96 = 436
Multiply by 2: 436 × 2 = 872 in²

✔ Correct as written on paper.

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Problem 2: Cylinder

Radius = 14 mm, Height = 29 mm

Formula for cylinder surface area:
SA = 2πr² + 2πrh
(That’s two circles on top/bottom + side rectangle)

Use π ≈ 3.14

First part (top and bottom):
2 × π × r² = 2 × 3.14 × 14² = 2 × 3.14 × 196 = 6.28 × 196 = 1230.88

Second part (side):
2 × π × r × h = 2 × 3.14 × 14 × 29
= 6.28 × 14 = 87.92; then 87.92 × 29 = let’s compute:

87.92 × 30 = 2637.6 → minus 87.92 = 2549.68

Total SA = 1230.88 + 2549.68 = 3780.56 mm²

Round to nearest hundredth → already is.

But wait — student wrote “1204π” which is about 3780.56? Let’s check:

If they used exact π:
2πr(r + h) = 2π×14×(14+29) = 28π×43 = 1204π → yes!

So if we use π = 3.1416, 1204 × 3.1416 ≈ ?

1200 × 3.1416 = 3769.92
4 × 3.1416 = 12.5664
Total ≈ 3782.4864 → rounds to 3782.49 mm²

Wait — earlier I got 3780.56 with π=3.14. Which is correct?

Actually, better to use more precise π.

Let me recalculate with π = 3.1415926535...

2πr² = 2 × π × 196 = 392π
2πrh = 2 × π × 14 × 29 = 812π
Total = (392 + 812)π = 1204π

Now 1204 × π = 1204 × 3.1415926535 ≈ ?

Compute:

1200 × π = 3769.911184
4 × π = 12.566370614
Sum = 3782.477555 → round to nearest hundredth → 3782.48 mm²

Student had “1204π” which is exact, but question says “round to nearest hundredth when necessary”, so we should give decimal.

✔ Final answer: 3782.48 mm²

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Problem 3: Triangular Prism

Base triangle: base = 8 cm, height = 3 cm, sides = 11 cm, 6 cm, 13 cm? Wait — diagram shows a triangular prism with:

- Triangle base: 8 cm, height 3 cm → area of one triangle = (1/2)*8*3 = 12 cm² → two triangles = 24 cm²
- Three rectangular faces:
- One is 8 cm × 6 cm? Wait — need to match edges.

Looking at diagram: It seems the triangular base has sides 8 cm, 6 cm, and 13 cm? But 8+6=14 >13, okay.

Actually, from drawing: The triangle has base 8 cm, height 3 cm (so area 12), and the three rectangles are along the three sides of the triangle, each with height (length of prism) = 6 cm? Wait — label says “6 cm” on the side edge.

Wait — actually, looking again: The prism has triangular ends with base 8 cm, height 3 cm, and the other two sides labeled 11 cm and 13 cm? That doesn’t make sense because if base is 8 and height is 3, the other sides can’t be 11 and 13 unless it’s not right triangle.

Wait — perhaps the 3 cm is the height of the triangle, and the sides are 8 cm (base), and the other two sides are given as 11 cm and 13 cm? But that would mean perimeter = 8+11+13=32 cm, times depth 6 cm? But depth is labeled as 6 cm? Actually, in diagram, the length of the prism (distance between triangles) is 6 cm.

Standard formula for triangular prism SA:

SA = 2 × (area of triangle) + (perimeter of triangle) × (height of prism)

Area of triangle = (1/2) × base × height = (1/2) × 8 × 3 = 12 cm² → two triangles = 24 cm²

Perimeter of triangle = sum of three sides. From diagram: sides are 8 cm, 11 cm, and 13 cm? But 8+11+13=32 cm? Then lateral area = 32 × 6 = 192 cm²

Total SA = 24 + 192 = 216 cm²

But student wrote: 3×11 + 3×6 + 3×8 + 2×(1/2×8×3) → that’s wrong. They used 3 instead of 6 for the prism height? And multiplied each side by 3? That’s incorrect.

Correct: Each rectangular face is side_of_triangle × length_of_prism = side × 6

So:

- Rectangle 1: 8 × 6 = 48
- Rectangle 2: 11 × 6 = 66
- Rectangle 3: 13 × 6 = 78
Sum rectangles = 48+66+78 = 192

Triangles: 2 × 12 = 24

Total = 192 + 24 = 216 cm²

✔ So student made mistake — used 3 instead of 6 for prism height.

Final Answer: 216 cm²

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Problem 4: Trapezoidal Prism

Trapezoid bases: 8 ft and 19 ft, height of trapezoid = 5 ft? Wait — diagram shows:

Top base = 8 ft, bottom base = 19 ft, height of trapezoid = 5 ft? But also there’s a slant side labeled 12.1 ft and another 14 ft? And the length of the prism is 12.1 ft? Wait — confusing.

Actually, looking: The trapezoid has parallel sides 8 ft and 19 ft, and the non-parallel sides are 12.1 ft and 14 ft? And the height of the trapezoid (distance between parallels) is 5 ft? Yes, marked with perpendicular line.

Length of prism (depth) = 12.1 ft? Or is 12.1 ft a side? Diagram shows "12.1 ft" on the left vertical edge — probably the length of the prism.

Actually, standard interpretation: The trapezoid is the base, with:

- Parallel sides: 8 ft and 19 ft
- Height of trapezoid: 5 ft (given with right angle mark)
- Non-parallel sides: 12.1 ft and 14 ft? But 12.1 ft is labeled on the side edge of the prism — likely the length of the prism.

Wait — re-examining: In the diagram, the trapezoid has:

- Top: 8 ft
- Bottom: 19 ft
- Left leg: 12.1 ft? But that might be the length of the prism.
Actually, no — the 12.1 ft is written on the left face, which is a rectangle — so likely the length of the prism is 12.1 ft.

And the trapezoid has height 5 ft (marked inside), and the two non-parallel sides are not given numerically except one is 14 ft? Wait — diagram shows "14 ft" on the right slanted side of the trapezoid.

So trapezoid sides: top 8, bottom 19, left side ? , right side 14 ft, height 5 ft.

To find area of trapezoid: A = (1/2)(b1 + b2)h = (1/2)(8+19)*5 = (1/2)(27)*5 = 13.5 * 5 = 67.5 ft² → two trapezoids = 135 ft²

Now lateral faces: four rectangles? No — trapezoidal prism has 4 lateral faces: one for each side of the trapezoid.

Sides of trapezoid:

- Top: 8 ft → rectangle: 8 × length_of_prism
- Bottom: 19 ft → rectangle: 19 × length_of_prism
- Left side: unknown? But diagram shows "12.1 ft" on the left vertical edge — that must be the length of the prism.
- Right side: 14 ft → rectangle: 14 × length_of_prism

What about the left side of the trapezoid? Not given numerically. But since height is 5 ft, and difference in bases is 19-8=11 ft, and if it's a right trapezoid? The 5 ft height is drawn perpendicular to the bases, and the left side is vertical? If so, then left side = 5 ft? But diagram labels "12.1 ft" on the left edge of the prism — which is the length, not the side.

I think the 12.1 ft is the length of the prism (the distance between the two trapezoidal bases).

Then the four lateral faces are:

1. Top: 8 ft × 12.1 ft = 96.8 ft²
2. Bottom: 19 ft × 12.1 ft = 229.9 ft²
3. Right slanted side: 14 ft × 12.1 ft = 169.4 ft²
4. Left side: what is its length? Since the trapezoid has height 5 ft, and assuming the left side is perpendicular (as drawn with right angle), then left side = 5 ft? But 5 ft is the height, not the side length — if it's perpendicular, then yes, the left side is 5 ft long.

In the diagram, there is a right angle symbol between the bottom base and the left side, meaning the left side is perpendicular to the bases, so it is the height, 5 ft.

So left side = 5 ft → rectangle: 5 × 12.1 = 60.5 ft²

Now sum lateral areas: 96.8 + 229.9 + 169.4 + 60.5 = let's add:

96.8 + 229.9 = 326.7
169.4 + 60.5 = 229.9
Total lateral = 326.7 + 229.9 = 556.6 ft²

Plus two trapezoids: 2 × 67.5 = 135 ft²

Total SA = 556.6 + 135 = 691.6 ft²

But student didn't write anything — so we'll go with this.

Is the left side really 5 ft? In a trapezoid, if one leg is perpendicular, it is equal to the height, yes.

So ✔ 691.6 ft²

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Problem 5: Square Prism (Rectangular Prism with square base)

Base: 9 m × 9 m, height = 13.5 m

SA = 2(lw + lh + wh) = 2(9×9 + 9×13.5 + 9×13.5)

Compute:

9×9 = 81
9×13.5 = 121.5
Another 9×13.5 = 121.5

Sum inside: 81 + 121.5 + 121.5 = 324

Times 2: 648 m²

Student has 648 m² — correct.

Note: Student wrote "2(9)(9) + 2(13.5)(9) + 10.5(9)(2)" — that last term is wrong; should be 2(9)(13.5) for the other pair, but they have 10.5 which is typo? Anyway, final number is correct.

✔ 648 m²

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Problem 6: Triangular Prism (Equilateral?)

Triangle sides: all 15 yd? Diagram shows equilateral triangle with side 15 yd, and height of triangle is 13 yd? Marked with perpendicular from vertex to base.

Length of prism = 24 yd

First, area of one triangle: (1/2) × base × height = (1/2) × 15 × 13 = 97.5 yd² → two triangles = 195 yd²

Lateral faces: three rectangles, each 15 yd × 24 yd = 360 yd² each? No — each rectangle is side × length = 15 × 24 = 360, and there are three such rectangles.

So lateral area = 3 × 360 = 1080 yd²

Total SA = 195 + 1080 = 1275 yd²

Student didn't write — so this is answer.

Note: Is the height 13 yd accurate for equilateral triangle? For equilateral triangle side s, height = (√3/2)s ≈ 0.866×15≈12.99, so 13 is approximate — fine for this context.

✔ 1275 yd²

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Problem 7: Cylinder

Diameter = 22 in → radius = 11 in, height = 14 in

SA = 2πr² + 2πrh = 2πr(r + h)

r = 11, h = 14

So r + h = 25

2πr(r+h) = 2π×11×25 = 550π

Now compute 550 × π ≈ 550 × 3.1415926535 = ?

500×π = 1570.79632679
50×π = 157.079632679
Sum = 1727.87595947 → round to nearest hundredth → 1727.88 in²

Student didn't write — so this is answer.

✔ 1727.88 in²

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Problem 8: Rectangular Prism

Dimensions: 23 cm, 20 cm, 4.5 cm

SA = 2(lw + lh + wh)

Set l=23, w=20, h=4.5

lw = 23×20 = 460
lh = 23×4.5 = 103.5
wh = 20×4.5 = 90

Sum = 460 + 103.5 + 90 = 653.5

Times 2 = 1307 cm²

Student didn't write — so this is answer.

✔ 1307 cm²

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## Now, compiling all answers:

1. 872 in²
2. 3782.48 mm²
3. 216 cm²
4. 691.6 ft²
5. 648 m²
6. 1275 yd²
7. 1727.88 in²
8. 1307 cm²

Note: For problem 4, I assumed the left side of trapezoid is 5 ft (since perpendicular). If not, we’d need more info, but based on diagram, it’s reasonable.

Also, for problem 3, corrected student’s error.

All calculations verified.

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Final Answer:
1. 872 in²
2. 3782.48 mm²
3. 216 cm²
4. 691.6 ft²
5. 648 m²
6. 1275 yd²
7. 1727.88 in²
8. 1307 cm²