worksheet-solid-figures- ... - Free Printable

Let’s solve each problem one by one. We’ll calculate the surface area of each solid figure using the correct formulas, and round to the nearest hundredth if needed.

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1. Rectangular Prism (13m x 14m x 17m)
Surface Area = 2(lw + lh + wh)
= 2(13×14 + 13×17 + 14×17)
= 2(182 + 221 + 238)
= 2(641) = 1282 m²

✔ Matches given answer.

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2. Square Pyramid (base 17 cm, slant height 10 cm)
Base area = 17 × 17 = 289 cm²
Lateral area = 4 × (½ × base × slant) = 4 × (½ × 17 × 10) = 4 × 85 = 340 cm²
Total SA = 289 + 340 = 629 cm²

Wait — that doesn’t match the given 1219.41 cm². Let me check again.

Actually, maybe it’s not a square pyramid? The diagram shows a pentagonal base? Wait — no, looking at the image description: it says “17 cm” on the base and “10 cm” as slant height. But if it’s a *square* pyramid with side 17 and slant 10, then lateral area is 4*(0.5*17*10)=340, plus base 289 → 629.

But the answer key says 1219.41 — that suggests maybe it’s a different shape or I misread.

Wait — perhaps the 10 cm is NOT the slant height but the edge from apex to base corner? That would make it a triangular face with sides 17, 10, 10? No, that doesn’t work.

Alternatively, maybe it’s a regular pyramid with base perimeter and apothem? Or perhaps the 10 cm is the height, not slant?

Let me recalculate assuming 10 cm is the vertical height.

Then we need slant height: for square pyramid, slant = √(height² + (side/2)²) = √(10² + 8.5²) = √(100 + 72.25) = √172.25 ≈ 13.124

Then lateral area = 4 × (½ × 17 × 13.124) ≈ 4 × 111.554 ≈ 446.216
Base = 289
Total ≈ 735.216 — still not 1219.

Hmm. Maybe it’s not a square pyramid? The diagram might show a hexagonal or other base? But label says “17 cm” — probably side length.

Wait — perhaps the 10 cm is the slant height, but the base is not square? If it’s a triangular pyramid? No, diagram looks like square base.

Alternatively, maybe the 17 cm is the diagonal? Unlikely.

Let me try another approach: perhaps the formula used was for total surface including all faces, and they calculated each triangle separately.

If base is square 17x17, and each triangular face has base 17 and height 10, then each triangle area = 0.5*17*10 = 85, times 4 = 340, plus base 289 = 629.

But 1219.41 is almost double — maybe they forgot to divide by 2 in triangle area? 4*17*10 = 680, plus 289 = 969 — still not.

Wait — what if the 10 cm is the edge length from apex to base vertex? Then we have an isosceles triangle with sides 10, 10, 17.

Area of one triangle: use Heron’s formula.

s = (10+10+17)/2 = 18.5
Area = √[18.5(18.5-10)(18.5-10)(18.5-17)] = √[18.5×8.5×8.5×1.5]
Calculate: 18.5×1.5 = 27.75; 8.5×8.5=72.25; so 27.75×72.25 ≈ let's compute:

27.75 × 72.25 = (28 - 0.25)(72 + 0.25) ≈ better to do direct:

27.75 × 72.25 = 27.75 × 70 + 27.75 × 2.25 = 1942.5 + 62.4375 = 2004.9375

√2004.9375 ≈ 44.78

So one triangle ≈ 44.78, four triangles ≈ 179.12, plus base 289 = 468.12 — still not.

This is confusing. Perhaps the figure is not a square pyramid? Looking back at user input: "17 cm" and "10 cm" — maybe it's a cone? No, it's drawn as pyramid.

Wait — perhaps the 17 cm is the perimeter? Unlikely.

Another idea: maybe it's a rectangular pyramid? But only one dimension given.

I think there might be a mistake in my assumption. Let me look at the answer: 1219.41 cm².

What if the base is 17 cm per side, and the slant height is such that... or perhaps the 10 cm is the height, and we need to find slant.

Earlier I did: slant = √(10² + 8.5²) = √172.25 ≈ 13.124

Then lateral area = 4 * 0.5 * 17 * 13.124 = 2 * 17 * 13.124 = 34 * 13.124 = let's calculate: 34*13 = 442, 34*0.124=4.216, total 446.216

Plus base 289 = 735.216 — not matching.

Perhaps the base is not included? But surface area usually includes all faces.

Or maybe it's a different shape. Let me skip and come back.

Actually, upon second thought, perhaps the "10 cm" is the slant height, and the base is 17 cm, but it's a triangular pyramid? No, diagram shows square base.

I recall that sometimes in worksheets, they might have errors, but let's assume the given answer is correct and see how they got it.

1219.41 - 289 = 930.41 for lateral area.

930.41 / 4 = 232.6025 per triangle.

0.5 * 17 * h = 232.6025 => h = 232.6025 * 2 / 17 ≈ 465.205 / 17 ≈ 27.365 — which is not 10.

So not matching.

Perhaps the 17 cm is not the side but the diagonal? If diagonal is 17, then side = 17/√2 ≈ 12.02, area = (12.02)^2 ≈ 144.48, then lateral area would need to be 1219.41 - 144.48 = 1074.93, divided by 4 = 268.73 per triangle, then 0.5*12.02*h = 268.73 => h = 268.73 *2 /12.02 ≈ 537.46/12.02 ≈ 44.7 — not 10.

This is not working. Let me move to next problems and return.

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3. Triangular Pyramid (Tetrahedron?) with edges 7 yd, 9 yd, 16 yd?
The diagram shows a triangular pyramid with base sides 7, 9, and another side? And height 16 yd? But 16 yd is labeled on the edge.

Typically, for a triangular pyramid, if it's irregular, we need more info. But here, perhaps it's a tetrahedron with given faces.

The answer is 322.01 yd².

Assume it's a pyramid with triangular base and three triangular faces.

Suppose the base is a triangle with sides 7, 9, and let's say the third side is not given, but perhaps it's a right triangle? 7-9-? Not right.

Use Heron's formula for each face.

But we don't know all sides.

Perhaps the 16 yd is the height of the pyramid, not an edge.

This is ambiguous. Let's assume the base is a triangle with sides 7, 9, and the included angle or something.

Perhaps it's a regular tetrahedron? But sides are different.

Another idea: perhaps the 16 yd is the slant height for the faces.

I think for now, since the answer is given, and this is a worksheet, I'll trust the provided answers for verification, but for solving, I need to calculate correctly.

Let's do the ones I can.

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4. Square Pyramid (base 10 ft, slant height 11 ft)
Base area = 10×10 = 100 ft²
Lateral area = 4 × (½ × 10 × 11) = 4 × 55 = 220 ft²
Total SA = 100 + 220 = 320 ft²

But given answer is 195.8 ft² — that's less than base alone? Impossible.

195.8 is less than 100? No, 195.8 > 100, but 320 is larger.

Perhaps the 11 ft is the vertical height, not slant.

Then slant height = √(11² + 5²) = √(121 + 25) = √146 ≈ 12.083

Lateral area = 4 × 0.5 × 10 × 12.083 = 2 × 10 × 12.083 = 241.66

Total SA = 100 + 241.66 = 341.66 — not 195.8.

If they forgot the base? Lateral only 241.66 — not 195.8.

Perhaps it's a different shape. Diagram shows square pyramid.

Another possibility: the 10 ft is the diagonal of the base? Then side = 10/√2 ≈ 7.07, area = 50, slant height if 11 is vertical, slant = √(11² + (7.07/2)^2) = √(121 + 12.5) = √133.5 ≈ 11.55, lateral area = 4*0.5*7.07*11.55 ≈ 2*7.07*11.55 ≈ 163.3, total 50+163.3=213.3 — close to 195.8? Not really.

195.8 might be for lateral only with different calculation.

Let's calculate what would give 195.8.

Suppose lateral area is 195.8, then for four triangles, each 48.95, with base 10, then 0.5*10*h = 48.95 => h = 9.79 — not 11.

I think there might be a mistake in the worksheet or my interpretation.

Perhaps the 11 ft is the edge, not height or slant.

For square pyramid, if edge from apex to base corner is 11, and base side 10, then distance from center to corner is (10√2)/2 = 5√2 ≈ 7.07, then height = √(11² - 7.07²) = √(121 - 50) = √71 ≈ 8.43, then slant height = √(8.43² + 5²) = √(71 + 25) = √96 ≈ 9.8, then lateral area = 4*0.5*10*9.8 = 196, plus base 100 = 296 — not 195.8.

If they want only lateral area, 196 is close to 195.8 — perhaps rounding.

√96 = 4√6 ≈ 4*2.449 = 9.796, then 4*0.5*10*9.796 = 2*10*9.796 = 195.92 — ah! Approximately 195.92, rounded to 195.8? 195.92 is closer to 195.9, but perhaps they have 195.8.

In the answer it's 195.8 ft², and if it's lateral area only, then yes.

But typically surface area includes base. However, in some contexts, "surface area" for pyramids might mean lateral, but usually not.

Looking at other problems, for example, the first one includes all faces.

For this one, if we assume the 11 ft is the edge length from apex to base vertex, then as above, slant height is √(h^2 + (s/2)^2), but h = √(edge^2 - (diagonal/2)^2) = √(11^2 - (5√2)^2) = √(121 - 50) = √71

Then slant height for the face is the distance from apex to midpoint of base side, which is √(h^2 + (s/2)^2) = √(71 + 25) = √96 = 4√6 ≈ 9.798

Then area of one triangular face = 0.5 * base * slant = 0.5 * 10 * 9.798 = 48.99

Four faces = 195.96 ≈ 196.0, but given is 195.8 — close, perhaps rounding difference.

Maybe they used exact values.

√96 = 4√6, area per face = 0.5 * 10 * 4√6 = 20√6

Four faces = 80√6

√6 ≈ 2.44949, 80*2.44949 = 195.9592, which rounds to 195.96, but the answer is 195.8 — perhaps a typo, or different interpretation.

In the worksheet, it's written as 195.8, so maybe they have a different value.

Perhaps the 11 ft is the slant height, but then why is answer small.

Another possibility: the base is not included, and they calculated lateral area as 195.8, which matches our calculation if 11 is the edge.

So for this problem, if we take the edge from apex to base corner as 11 ft, base side 10 ft, then lateral surface area is 80√6 ≈ 195.96, and if rounded to nearest tenth, 196.0, but given 195.8 — perhaps calculation error in worksheet.

We'll proceed with the calculation as per standard.

But to match the worksheet, perhaps for this one, they intend lateral area only, and with their numbers.

Let's move on and come back.

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5. Cylinder (diameter 7 mm, height 6 mm)
Radius r = 7/2 = 3.5 mm
Surface Area = 2πr² + 2πrh = 2πr(r + h) = 2π*3.5*(3.5 + 6) = 7π*9.5 = 66.5π
π≈3.1416, 66.5*3.1416 ≈ let's calculate: 66.5*3 = 199.5, 66.5*0.1416≈9.4164, total 208.9164 ≈ 208.92 mm² — matches given.

Good.

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6. Trapezoidal Prism (bases 12 yd and 16 yd, height of trapezoid 7 yd, length of prism 15 yd)
First, area of trapezoidal base = (b1 + b2)/2 * h = (12+16)/2 * 7 = 14*7 = 98 yd²
Two bases: 2*98 = 196 yd²

Now lateral area: the prism has 4 rectangular faces.

The two parallel sides of trapezoid are 12 and 16, and the non-parallel sides are not given, but in the diagram, it shows the height of trapezoid is 7 yd, and the length of prism is 15 yd.

To find the lengths of the non-parallel sides, we need more info. In a trapezoid with bases 12 and 16, height 7, the difference in bases is 4 yd, so if it's isosceles, each overhang is 2 yd, so leg = √(2² + 7²) = √(4+49) = √53 ≈ 7.28 yd.

Then the lateral faces are:
- Rectangle for base 12: 12 * 15 = 180 yd²
- Rectangle for base 16: 16 * 15 = 240 yd²
- Two rectangles for legs: each 7.28 * 15 ≈ 109.2 yd², so two = 218.4 yd²

Total lateral area = 180 + 240 + 218.4 = 638.4 yd²

Total SA = bases + lateral = 196 + 638.4 = 834.4 yd²

But given answer is 621 yd² — not matching.

Perhaps the 7 yd is not the height of the trapezoid, but something else.

In the diagram, it shows "7 yd" as the height of the trapezoid, and "15 yd" as the length, and bases 12 and 16.

Maybe the non-parallel sides are given? In the text, it says "7 yd" and "15 yd", but in the description, it's "7 yd" for height, and perhaps the legs are not needed if we have the perimeter.

Another way: perhaps the lateral area is perimeter of base times height of prism.

Perimeter of trapezoid = 12 + 16 + leg1 + leg2.

If isosceles, legs = √((16-12)/2)^2 + 7^2) = √(2^2 + 7^2) = √53 ≈7.28, so perimeter = 12+16+7.28+7.28 = 42.56 yd

Lateral area = 42.56 * 15 = 638.4 yd², same as before.

Total SA = 2*98 + 638.4 = 196 + 638.4 = 834.4

But given is 621 — much smaller.

Perhaps the 7 yd is the length of the leg, not the height.

Let me read the user input: "12yd, 16yd, 7yd, 15yd" — and in diagram, 7yd is likely the height of the trapezoid.

Maybe for surface area, they only want lateral, but 638.4 vs 621 — close but not exact.

621 / 15 = 41.4, so perimeter would be 41.4, but 12+16=28, so legs sum to 13.4, each 6.7, then height would be √(6.7^2 - 2^2) = √(44.89 - 4) = √40.89 ≈6.4, not 7.

Not matching.

Perhaps the 7 yd is the height of the prism, and 15 yd is something else — but in diagram, 15 yd is the length.

I think there might be a mistake, or perhaps the trapezoid is right-angled.

Assume it's a right trapezoid with bases 12 and 16, height 7, then the leg perpendicular is 7, and the other leg is √((16-12)^2 + 7^2) = √(16+49) = √65 ≈8.06

Then perimeter = 12+16+7+8.06 = 43.06

Lateral area = 43.06*15 = 645.9

Bases 2*98=196, total 841.9 — still not 621.

Perhaps the "7 yd" is the length of the non-parallel side, and height is not given.

This is problematic.

Let's look at the answer 621 yd².

Suppose lateral area is P * l, and bases are included.

2* [(12+16)/2 * h] + P*15 = 621

But h is unknown.

Perhaps in the diagram, the 7 yd is the height, and the legs are equal, but then as before.

Another idea: perhaps the 15 yd is not the length, but the height of the trapezoid, and 7 yd is the length of the prism.

Let me try that.

Suppose the trapezoid has bases 12 and 16, height 15 yd (instead of 7), then area = (12+16)/2 * 15 = 14*15 = 210 yd², two bases 420 yd².

Then legs: if isosceles, overhang 2 yd, leg = √(2^2 + 15^2) = √(4+225) = √229 ≈15.132

Perimeter = 12+16+15.132+15.132 = 58.264

Lateral area = 58.264 * 7 = 407.848 (if 7 yd is length)

Total SA = 420 + 407.848 = 827.848 — not 621.

If 7 yd is the height of trapezoid, and 15 yd is length, but perhaps the bases are not both included or something.

Perhaps for this prism, the surface area is calculated as 2*base + lateral, but with different values.

Let's calculate what would give 621.

Suppose the base area is B, lateral L, 2B + L = 621.

B = (12+16)/2 * h = 14h

L = P * 15, P = 12+16+2*leg, leg = √((2)^2 + h^2) = √(4+h^2)

So 2*14h + 15*(28 + 2√(4+h^2)) = 621

28h + 420 + 30√(4+h^2) = 621

28h + 30√(4+h^2) = 201

Let u = h, 28u + 30√(4+u^2) = 201

Divide by 2: 14u + 15√(4+u^2) = 100.5

Let v = √(4+u^2), then 14u + 15v = 100.5, and v^2 = 4+u^2

From 14u + 15v = 100.5, v = (100.5 - 14u)/15

Then [(100.5 - 14u)/15]^2 = 4 + u^2

Multiply both sides by 225: (100.5 - 14u)^2 = 225(4 + u^2)

Calculate 100.5^2 = 10100.25, 2*100.5*14u = 2814u, (14u)^2 = 196u^2

Left: 10100.25 - 2814u + 196u^2

Right: 900 + 225u^2

So 10100.25 - 2814u + 196u^2 = 900 + 225u^2

Bring all to left: 10100.25 - 2814u + 196u^2 - 900 - 225u^2 = 0

-29u^2 - 2814u + 9200.25 = 0

Multiply by -1: 29u^2 + 2814u - 9200.25 = 0

Discriminant d = b^2 - 4ac = 2814^2 - 4*29*(-9200.25) wait no, a=29, b=2814, c= -9200.25

d = 2814^2 - 4*29*(-9200.25) = 2814^2 + 4*29*9200.25

This is messy, and likely not integer.

Perhaps the 7 yd is the length of the leg, and height is unknown.

I think for the sake of time, since this is a worksheet with answers provided, and my job is to verify or solve, but the user wants me to solve, so I should calculate correctly.

Perhaps in the diagram, the 7 yd is the height of the trapezoid, and the non-parallel sides are given as 7 yd each? But that would not be possible with bases 12 and 16, because the difference is 4, so if legs are 7, then height = √(7^2 - 2^2) = √(49-4) = √45 = 3√5 ≈6.708, then area = (12+16)/2 * 6.708 = 14*6.708 = 93.912, two bases 187.824

Perimeter = 12+16+7+7 = 42

Lateral area = 42 * 15 = 630

Total SA = 187.824 + 630 = 817.824 — not 621.

If the 7 yd is the height, and legs are not 7, but in the text it's "7 yd" for the height.

I recall that in some worksheets, for trapezoidal prism, they might give the dimensions differently.

Perhaps the 15 yd is the height of the trapezoid, and 7 yd is the length, but then as before.

Let's try: bases 12, 16, height of trapezoid 15, then area = 14*15 = 210, two bases 420.

Legs: if isosceles, leg = √(2^2 + 15^2) = √229 ≈15.132, perimeter = 12+16+15.132+15.132 = 58.264

Lateral area = 58.264 * 7 = 407.848

Total 420 + 407.848 = 827.848 — not 621.

Perhaps the "7 yd" is the length of the prism, and "15 yd" is the height of the trapezoid, but same thing.

Another idea: perhaps the 7 yd is the distance between the bases or something.

I think I need to accept that for this one, with the given numbers, and the answer 621, perhaps they have a different interpretation.

Maybe the surface area is only the lateral area, and 621 / 15 = 41.4, so perimeter 41.4, with bases 12 and 16, sum 28, so legs sum 13.4, each 6.7, then if isosceles, height = √(6.7^2 - 2^2) = √(44.89 - 4) = √40.89 = 6.4, then base area = 14*6.4 = 89.6, but if only lateral, 621, which is 41.4*15, so perhaps they mean lateral surface area only.

In many contexts, "surface area" for prisms includes all faces, but perhaps for this worksheet, for some figures, they mean lateral.

For the cylinder, they included top and bottom, since 208.92 includes 2πr^2.

For the rectangular prism, includes all.

For this, perhaps not.

But 621 is given, and if lateral area is P* l = 621, with l=15, P=41.4, which is reasonable if legs are 6.7 each.

But in the diagram, it's labeled as 7 yd for the height, so likely not.

Perhaps the 7 yd is the length of the leg, and height is 7 yd, but then for bases 12 and 16, if leg is 7, height must be less.

I think for the purpose of this exercise, since the worksheet has answers, and my role is to help the student, I'll provide the correct calculations based on standard formulas, and note discrepancies.

But to save time, let's do the others.

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**7. Rectangular Prism (5 ft x 5 ft x 5 ft)? Wait, it's a cube? No, dimensions 5 ft, 5 ft, and 5 ft? But in diagram, it's a rectangular prism with square base 5x5, height 5 ft? Then it's a cube.

Surface area = 6 * s^2 = 6*25 = 150 ft², but given 211.02 — not matching.

Dimensions: from user input, "5 ft" for width, "5 ft" for depth, "5 ft" for height? But in the text, it's "5 ft" for the base sides, and "5 ft" for height, so yes, cube.

But 6*25=150, not 211.02.

Perhaps it's not a cube; maybe the height is different.

In the user input: "5 ft" for the base, and "5 ft" for the height, but perhaps the base is 5x5, height is something else.

In the description: "5 ft" for the side, and "5 ft" for the height, but in the diagram, it might be different.

Perhaps it's a rectangular prism with dimensions 5, 5, and h, but h is not given.

In the text: "5 ft" for the base sides, and "5 ft" for the height, so likely 5x5x5.

But 150 ≠ 211.02.

Unless the 5 ft is not the side, but something else.

Perhaps it's a different shape. Diagram shows a rectangular prism with square base, but height may be different.

In the user input, it's "5 ft" for the width, "5 ft" for the depth, and "5 ft" for the height, but perhaps the "5 ft" for height is not correct.

Let's look at the answer 211.02.

For a rectangular prism with sides a,b,c, SA = 2(ab+bc+ca)

If a=b=5, then SA = 2(25 + 5c + 5c) = 2(25 + 10c) = 50 + 20c

Set equal to 211.02: 50 + 20c = 211.02, 20c = 161.02, c = 8.051 — not 5.

So probably the height is not 5 ft.

In the diagram, perhaps the height is labeled as 5 ft, but in reality, for the calculation, it's different.

Perhaps the 5 ft is the diagonal or something.

I think there might be a mistake in the worksheet or my reading.

For the sake of proceeding, let's assume the dimensions are given, and calculate.

Perhaps "5 ft" for the base, and the height is 5 ft, but it's not a cube; wait, if base is 5x5, height 5, it is a cube.

Another idea: perhaps the "5 ft" is the length of the edge, but for a different orientation.

I recall that in some diagrams, they might have the height as 5 ft, but the base is not 5x5.

In the user input, it's "5 ft" for the width, "5 ft" for the depth, so likely square base.

Perhaps the 5 ft is the side, but the height is different, but not specified.

In the text, it's "5 ft" for the base sides, and "5 ft" for the height, so I think it's intended to be 5x5x5, but answer is wrong, or perhaps it's 5x5x8 or something.

Let's calculate what height gives 211.02.

As above, 50 + 20c = 211.02, c = 8.051, so approximately 8.05 ft.

But in the diagram, it's labeled as 5 ft, so likely error.

Perhaps the 5 ft is the diagonal of the base.

If base diagonal is 5 ft, then side = 5/√2 ≈3.535, area = 12.5, then SA = 2*12.5 + 4*3.535*h = 25 + 14.14h

Set to 211.02, 14.14h = 186.02, h = 13.15 — not 5.

Not matching.

I think for this, I'll skip and do the ones I can.

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8. Cube (8 cm side)
SA = 6 * s^2 = 6*64 = 384 cm² — matches given.

Good.

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9. Cone (radius 2.5 in, slant height 15 in)
SA = πr² + πr l = πr(r + l) = π*2.5*(2.5 + 15) = 2.5π*17.5 = 43.75π
π≈3.1416, 43.75*3.1416 ≈ 137.445, but given 1206.37 — way off.

1206.37 / π ≈ 384, so perhaps r(r+l) = 384, with r=2.5, 2.5*(2.5+l) = 384, 2.5+l = 153.6, l=151.1 — not 15.

Perhaps the 15 in is the height, not slant.

Then slant l = √(r^2 + h^2) = √(6.25 + 225) = √231.25 ≈ 15.207

SA = π*2.5*(2.5 + 15.207) = 2.5π*17.707 ≈ 2.5*3.1416*17.707 ≈ 7.854*17.707 ≈ 139.1 — not 1206.

1206.37 is large, so perhaps radius is larger.

If r=15, l=2.5, but that doesn't make sense.

Perhaps the 2.5 in is the diameter, so r=1.25 in.

Then SA = π*1.25*(1.25 + 15) = 1.25π*16.25 = 20.3125π ≈ 63.8 — not 1206.

1206.37 / π ≈ 384, so r(r+l) = 384.

If r=12, then 12*(12+l) = 384, 12+l = 32, l=20.

But in diagram, it's 2.5 and 15.

Perhaps the 15 in is the radius, and 2.5 in is the height or something.

Assume r=15, h=2.5, then l = √(225 + 6.25) = √231.25 ≈15.207, SA = π*15*(15 + 15.207) = 15π*30.207 ≈ 15*3.1416*30.207 ≈ 47.124*30.207 ≈ 1423 — not 1206.

If r=10, l=15, SA = π*10*(10+15) = 250π ≈ 785.4 — not 1206.

If r=12, l=15, SA = π*12*27 = 324π ≈ 1017.88 — close to 1206? 1017 vs 1206, not very.

324*3.1416=1017.8784, while 1206.37 / 3.1416 ≈ 384, as before.

So r(r+l) = 384.

Suppose r=16, then 16*(16+l) = 384, 16+l = 24, l=8.

But in diagram, it's 2.5 and 15.

Perhaps the 2.5 in is not the radius, but the diameter, and 15 in is slant, but then r=1.25, SA = π*1.25*(1.25+15) = 1.25*16.25*π = 20.3125π ≈ 63.8.

I think there might be a units error or something.

Another idea: perhaps the 2.5 in is the radius, but the 15 in is the height, and they want total surface area, but as calculated, it's small.

1206.37 is approximately 384*π, and 384 = 16*24, etc.

Perhaps for this cone, the radius is 12 in, slant 15 in, but 12*27*π = 324π ≈ 1017, not 1206.

1206 / π ≈ 384, and 384 = 16*24, or 12*32, etc.

Suppose r=12, then r+l = 32, l=20, SA = π*12*32 = 384π ≈ 1206.37 — yes! 384*3.1415926535 = let's calculate: 384*3 = 1152, 384*0.1415926535 ≈ 384*0.14 = 53.76, 384*0.0015926535≈0.611, total 53.76+0.611=54.371, so 1152+54.371=1206.371 — perfect.

So for the cone, radius is 12 in, slant height is 20 in? But in the diagram, it's labeled as 2.5 in and 15 in.

Perhaps the 2.5 in is a typo, and it's 12 in, and 15 in is not slant.

In the user input, it's "2.5in" and "15in", but for the cone, if r=12, l=20, but 20 is not given.

Perhaps the 15 in is the height, and r=12, then l = √(144 + 225) = √369 ≈19.209, SA = π*12*(12+19.209) = 12π*31.209 ≈ 12*3.1416*31.209 ≈ 37.6992*31.209 ≈ 1176.5 — not 1206.

With r=12, l=20, SA=384π≈1206.37, so likely the radius is 12 in, and slant height is 20 in, but in the diagram, it's labeled as 2.5 and 15, which is probably a mistake, or perhaps the 2.5 is for something else.

In the user input, for the cone, it's "2.5in" and "15in", but perhaps "2.5in" is the radius, and "15in" is the height, but then SA is small.

Perhaps "2.5in" is the diameter, so r=1.25, but then SA is small.

I think for this, we'll assume that the intended values are r=12 in, l=20 in, but since the diagram shows 2.5 and 15, perhaps it's a different scale.

To match the answer, for the cone, SA = πr(r+l) = 1206.37, so r(r+l) = 384.

With the given 2.5 and 15, if we take r=2.5, l=15, then r(r+l) = 2.5*17.5 = 43.75, SA=43.75π≈137.44, not 1206.

So likely, the 2.5 in is not the radius; perhaps it's the height, and 15 in is the radius.

Try r=15, h=2.5, then l = √(225 + 6.25) = √231.25 = 15.207, SA = π*15*(15+15.207) = 15π*30.207 = 453.105π ≈ 1423.5 — not 1206.

If r=10, h=15, l= √(100+225) = √325 = 5√13 ≈18.028, SA = π*10*(10+18.028) = 10π*28.028 = 280.28π ≈ 880.5 — not.

Only when r=12, l=20, it works.

Perhaps the 15 in is the slant height, and 2.5 in is not related, but in diagram, it's labeled.

I think for the sake of completing, I'll use the correct formula with given numbers, but since the answer is given, and for the student, I'll provide the calculation as per standard.

But to save time, let's do the last few.

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10. Sphere (radius 5 mm)
SA = 4πr² = 4π*25 = 100π ≈ 314.16 mm² — matches given.

Good.

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11. Square Pyramid (base 11 mm, slant height 13 mm)
Base area = 11*11 = 121 mm²
Lateral area = 4 * (0.5 * 11 * 13) = 2 * 11 * 13 = 286 mm²
Total SA = 121 + 286 = 407 mm²

But given answer is 380.14 mm² — not matching.

If 13 mm is the height, not slant.

Then slant = √(13^2 + (11/2)^2) = √(169 + 30.25) = √199.25 ≈14.115

Lateral area = 4 * 0.5 * 11 * 14.115 = 2 * 11 * 14.115 = 310.53

Total SA = 121 + 310.53 = 431.53 — not 380.14.

If they want only lateral area, 310.53, not 380.14.

380.14 - 121 = 259.14 for lateral, so per triangle 64.785, with base 11, 0.5*11*h = 64.785, h = 11.78 — not 13.

Perhaps the base is not included, and slant is 13, lateral area 286, not 380.

I think there might be a mistake.

Perhaps the 11 mm is the diagonal of the base.

If diagonal 11, side = 11/√2 ≈7.778, area = 60.5, then if slant 13, lateral area = 4*0.5*7.778*13 = 2*7.778*13 = 202.228, total 60.5+202.228=262.728 — not 380.

If r=11 for something else.

Another idea: perhaps it's a different pyramid.

Or perhaps the 13 mm is the edge.

Assume edge from apex to base corner is 13 mm, base side 11 mm.

Then distance from center to corner = (11√2)/2 = 5.5√2 ≈7.778

Height h = √(13^2 - 7.778^2) = √(169 - 60.5) = √108.5 ≈10.416

Then slant height for face = √(h^2 + (s/2)^2) = √(108.5 + 30.25) = √138.75 ≈11.78

Then lateral area = 4 * 0.5 * 11 * 11.78 = 2 * 11 * 11.78 = 259.16

Total SA = 121 + 259.16 = 380.16 — ah! Approximately 380.16, and given 380.14 — close, rounding difference.

So for this pyramid, the 13 mm is the edge length from apex to base vertex, not the slant height.

Similarly for earlier problems.

So for consistency, in the worksheet, for pyramids, the given "height" may be the edge length, not the slant or vertical height.

For problem 2, square pyramid with base 17 cm, edge 10 cm? But 10 cm is small for edge if base is 17.

Distance from center to corner for base 17 is (17√2)/2 = 8.5√2 ≈12.02, so if edge is 10, then height = √(10^2 - 12.02^2) = √(100 - 144.48) = √(-44.48) impossible.

So for problem 2, if base 17 cm, and edge 10 cm, it's impossible since 10 < 12.02.

Perhaps for problem 2, the 10 cm is the slant height, and they have a different calculation.

For problem 4, we had lateral area 195.96 when edge is 11, base 10.

For problem 11, with edge 13, base 11, SA=380.16≈380.14.

For problem 2, if base 17 cm, and edge e, then distance to corner d = 17/√2 * √2 /2 wait, d = (s√2)/2 = s/√2 = 17/1.4142 ≈12.02, so e > 12.02, but given 10<12.02, impossible.

So for problem 2, likely the 10 cm is the slant height, and they have a different answer, or perhaps it's a different shape.

Perhaps for problem 2, it's a cone, but drawn as pyramid.

I think for the student, I'll provide the correct calculations based on the labels, but since the worksheet has answers, and to match, I'll use the values that give the answers.

But for the response, I'll solve each with the given numbers and correct formulas.

To finalize, let's list the correct surface areas based on standard interpretation.

For the rectangular prism 13x14x17: SA = 2(13*14 + 13*17 + 14*17) = 2(182+221+238) = 2*641 = 1282 m² — correct.

For the square pyramid with base 17 cm, and if 10 cm is slant height, SA = base + lateral = 17^2 + 4*(0.5*17*10) = 289 + 340 = 629 cm², but given 1219.41, so perhaps 10 cm is not slant.

If 10 cm is the height, then slant = √(10^2 + (8.5)^2) = √(100+72.25) = √172.25 = 13.124, SA = 289 + 4*0.5*17*13.124 = 289 + 2*17*13.124 = 289 + 34*13.124 = 289 + 446.216 = 735.216 cm².

Still not 1219.41.

Perhaps the base is not square; maybe it's a different polygon.

I think for the sake of this response, since the user has the worksheet with answers, and my role is to help, I'll provide the final answers as per the worksheet, but with correct reasoning for the ones I can.

Perhaps for problem 2, the 17 cm is the perimeter or something.

Another idea: perhaps "17 cm" is the side, but it's a hexagonal pyramid or something, but unlikely.

I recall that 1219.41 / π ≈ 388.1, not nice.

1219.41 / 17 = 71.73, not helpful.

Perhaps it's a cone with r=17, l=10, but then SA = π*17*(17+10) = 17*27*π = 459π ≈ 1442, not 1219.

If r=10, l=17, SA = π*10*27 = 270π ≈ 848.23 — not.

I give up on problem 2 for now.

For the response, I'll box the answers as per the worksheet, since the student has them, and my job is to verify or explain, but the user said "solve the problem accurately", so I should calculate correctly.

Perhaps in the worksheet, for the square pyramid, the 10 cm is the slant height, and they have a calculation error, but for the student, I'll provide the correct one.

To move forward, let's assume for each figure, we use the given dimensions and correct formulas.

For the cone in problem 9, with r=2.5 in, l=15