Surface area and volume worksheet for composite solids with diagrams and calculation fields.

Let’s solve each shape one by one. We’ll find surface area and volume for each, rounding to the nearest tenth as instructed.

We’ll use these formulas:

Cylinder:
- Surface Area = 2πr² + 2πrh
- Volume = πr²h

Cone:
- Surface Area = πr² + πrℓ (where is slant height)
- Volume = (1/3)πr²h

Sphere:
- Surface Area = 4πr²
- Volume = (4/3)πr³

Cube:
- Surface Area = 6s²
- Volume = s³

Rectangular Prism:
- Surface Area = 2(lw + lh + wh)
- Volume = lwh

Composite Shapes: Break into parts, calculate separately, then add or subtract as needed.

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Shape ①: Cylinder with hemisphere on top (like a capsule)

Given:
- Radius r = 5 ft
- Height of cylinder part = 8 ft
- Hemisphere has same radius → total height = 8 + 5 = 13 ft? But diagram shows 8 ft for cylinder only — so we assume the 8 ft is just the cylindrical part.

Actually, looking at the diagram: it's a cylinder with a half-sphere on top. The 8 ft is the height of the cylinder. The hemisphere adds no extra height in calculation since we’re given radius.

But wait — surface area: we don’t include the base of the hemisphere (it’s attached), and we don’t include the top circle of the cylinder (covered by hemisphere). So:

Surface Area = lateral area of cylinder + area of bottom circle + curved surface of hemisphere

= 2πrh + πr² + 2πr²
= 2πrh + 3πr²

Wait — actually, standard way:

For a cylinder with a hemisphere on top (closed at bottom):

- Bottom: circle → πr²
- Side of cylinder: 2πrh
- Top: hemisphere curved surface → 2πr²

So total SA = πr² + 2πrh + 2πr² = 3πr² + 2πrh

Volume = volume of cylinder + volume of hemisphere
= πr²h + (2/3)πr³

Plug in r = 5, h = 8

SA = 3π(25) + 2π(5)(8) = 75π + 80π = 155π ≈ 155 * 3.1416 ≈ 486.9 ft²

Volume = π(25)(8) + (2/3)π(125) = 200π + (250/3)π = (600/3 + 250/3)π = (850/3)π ≈ 283.333 * 3.1416 ≈ 890.1 ft³

Wait — let me recalculate numerically:

π ≈ 3.1416

SA = 3 * π * 25 = 75π ≈ 235.62
2πrh = 2 * π * 5 * 8 = 80π ≈ 251.33
Total SA ≈ 235.62 + 251.33 = 486.95 → 487.0 ft²

Volume:
Cylinder: π*25*8 = 200π ≈ 628.32
Hemisphere: (2/3)*π*125 = (250/3)π ≈ 83.333*3.1416 ≈ 261.80
Total Vol ≈ 628.32 + 261.80 = 890.12 → 890.1 ft³

✔ Shape ①:
Surface Area = 487.0 ft²
Volume = 890.1 ft³

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Shape ②: Cone

Given:
- Radius r = 5 ft
- Height h = 10 ft
- Need slant height ℓ for surface area.

ℓ = √(r² + h²) = √(25 + 100) = √125 ≈ 11.1803 ft

Surface Area = πr² + πrℓ = π(25) + π(5)(11.1803) = 25π + 55.9015π = 80.9015π ≈ 80.9015 * 3.1416 ≈ 254.1 ft²

Volume = (1/3)πr²h = (1/3)π(25)(10) = (250/3)π ≈ 83.333 * 3.1416 ≈ 261.8 ft³

✔ Shape ②:
Surface Area = 254.1 ft²
Volume = 261.8 ft³

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Shape ③: Composite — cone on top of cylinder

Given:
- Both have radius r = 3 yd
- Cylinder height = 4 yd
- Cone height = 3 yd

First, slant height of cone: ℓ = √(3² + 3²) = √18 ≈ 4.2426 yd

Surface Area:
- Bottom of cylinder: πr²
- Side of cylinder: 2πrh_cyl
- Curved surface of cone: πrℓ
→ Do NOT include the top of cylinder or base of cone (they are joined)

So SA = πr² + 2πr*h_cyl + πr*ℓ
= π(9) + 2π(3)(4) + π(3)(4.2426)
= 9π + 24π + 12.7278π = 45.7278π ≈ 45.7278 * 3.1416 ≈ 143.7 yd²

Volume:
Cylinder: πr²h = π(9)(4) = 36π
Cone: (1/3)πr²h = (1/3)π(9)(3) = 9π
Total Vol = 45π ≈ 141.4 yd³

✔ Shape ③:
Surface Area = 143.7 yd²
Volume = 141.4 yd³

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Shape ④: Cube

Side = 5.5 cm

SA = 6 * (5.5)² = 6 * 30.25 = 181.5 cm²
Vol = (5.5)³ = 166.375 → 166.4 cm³

✔ Shape ④:
Surface Area = 181.5 cm²
Volume = 166.4 cm³

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Shape ⑤: Rectangular prism (box)

Dimensions: 4 cm × 6 cm × 5 cm

SA = 2(lw + lh + wh) = 2(4*6 + 4*5 + 6*5) = 2(24 + 20 + 30) = 2(74) = 148 cm²
Vol = 4*6*5 = 120 cm³

✔ Shape ⑤:
Surface Area = 148.0 cm²
Volume = 120.0 cm³

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Shape ⑥: Cylinder

Radius r = 3 in, Height h = 8 in

SA = 2πr² + 2πrh = 2π(9) + 2π(3)(8) = 18π + 48π = 66π ≈ 207.3 in²
Vol = πr²h = π(9)(8) = 72π ≈ 226.2 in³

✔ Shape ⑥:
Surface Area = 207.3 in²
Volume = 226.2 in³

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Shape ⑦: Rectangular prism

Dimensions: 10 m × 5 m × 4 m

SA = 2(10*5 + 10*4 + 5*4) = 2(50 + 40 + 20) = 2(110) = 220 m²
Vol = 10*5*4 = 200 m³

✔ Shape ⑦:
Surface Area = 220.0 m²
Volume = 200.0 m³

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Shape ⑧: Composite — small cube on top of larger cube

Large cube: side = 4 ft
Small cube: side = 2 ft

When you stack them, the bottom face of the small cube covers part of the top face of the large cube.

So total surface area:

Large cube SA = 6*(4)² = 96 ft²
Small cube SA = 6*(2)² = 24 ft²
But when stacked, two faces are hidden: bottom of small cube and the covered part of large cube’s top → both are 2x2 = 4 ft² each → total hidden = 8 ft²

So total SA = 96 + 24 - 8 = 112 ft²

Volume = 4³ + 2³ = 64 + 8 = 72 ft³

✔ Shape ⑧:
Surface Area = 112.0 ft²
Volume = 72.0 ft³

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Shape ⑨: L-shaped prism (composite rectangular prisms)

Break into two rectangles:

Option 1: Big rectangle minus missing corner.

Overall dimensions: length 10 m, width 6 m, but there’s a cutout.

Looking at diagram: it’s like a big box 10m long, 6m wide, 3m high, but with a 4m x 3m x 3m piece missing from one end? Wait — better to split into two parts.

From diagram: left part is 6m long, 3m wide, 3m high? Actually, labels:

It says: overall length 10m, depth 6m, height 3m. And there’s a notch: 4m along the length, 3m deep? Let me interpret.

Actually, common way: this is an L-shape in plan view.

Assume:

Part A: 6m (length) × 6m (width) × 3m (height) — but that doesn't fit.

Better: look at the numbers:

The figure has:

- Total length: 10 m
- Width: 6 m
- Height: 3 m
- There’s a step: from left, 6m long section full width, then next 4m only 3m wide? Or vice versa.

Actually, typical interpretation:

Imagine looking from top:

Left part: 6m long × 6m wide
Right part: 4m long × 3m wide (since total length 10m, and width reduces to 3m on right)

But height is uniform 3m.

So volumes:

Part 1: 6 × 6 × 3 = 108 m³
Part 2: 4 × 3 × 3 = 36 m³
Total Vol = 144 m³

Now surface area:

This is tricky. Better to think of external surfaces.

Since it’s a prism, we can compute perimeter of base times height, plus top and bottom areas.

Base shape (L-shape):

Area of base = area of big rectangle minus missing part? Or add two rectangles.

As above: 6x6 + 4x3 = 36 + 12 = 48 m² → so top and bottom together: 2 * 48 = 96 m²

Now lateral surface area: perimeter of base × height

Perimeter of L-shape:

Start from bottom-left: go right 6m, up 3m (because width drops), right 4m, up 3m? No.

Actually, if base is:

- From (0,0) to (6,6) — rectangle
- Then from (6,0) to (10,3) — another rectangle? That would overlap.

Standard L-shape for this problem:

Usually: total bounding box 10m x 6m, but missing a 4m x 3m rectangle from top-right or something.

Looking at common problems: often it's composed of two rectangles:

Rectangle A: 6m (len) x 6m (wid) x 3m (ht)
Rectangle B: 4m (len) x 3m (wid) x 3m (ht), attached to the side.

But if attached along the 3m side, then the combined shape has:

Length: 6 + 4 = 10m
Width: max(6,3) = 6m, but only 3m wide for the last 4m.

So base area = 6*6 + 4*3 = 36 + 12 = 48 m² — correct.

Perimeter: trace the outer edge.

Start at bottom-left corner:

- Right 6m (along bottom of first rect)
- Up 3m (to where second rect starts? No)

Actually, if the second rectangle is attached to the right side of the first, but only half height? I think I need to visualize differently.

Alternative approach: since height is constant 3m, surface area = 2 * base_area + lateral_area

Lateral_area = perimeter_of_base * height

So what is perimeter of the L-shaped base?

Assume the base looks like:

```
+-------+
| |
| A | 6m
| |
+---+---+
| B | 3m
+---+
4m
```

So total width: 6m (for A), and B is 3m wide, attached to the right side of A, but only covering the bottom 3m of A's right side.

So coordinates:

A: from (0,0) to (6,6)
B: from (6,0) to (10,3)

Then the outer boundary:

Start at (0,0):
→ right to (6,0) : 6m
↑ up to (6,3) : 3m (but this is internal? No, because B starts here)

Actually, from (0,0) to (10,0): but not straight.

Better path:

Start at (0,0):
- Right to (10,0)? No, because from x=6 to x=10, y only goes to 3.

Actually, the bottom edge is from (0,0) to (10,0) — 10m
Then up at x=10 to (10,3) — 3m
Then left to (6,3) — 4m
Then up to (6,6) — 3m
Then left to (0,6) — 6m
Then down to (0,0) — 6m

Is that correct? Let's see:

Points: (0,0) -> (10,0) -> (10,3) -> (6,3) -> (6,6) -> (0,6) -> (0,0)

Distances:

(0,0)-(10,0): 10
(10,0)-(10,3): 3
(10,3)-(6,3): 4
(6,3)-(6,6): 3
(6,6)-(0,6): 6
(0,6)-(0,0): 6

Sum: 10+3+4+3+6+6 = 32 m

Yes, perimeter = 32 m

So lateral surface area = 32 * 3 = 96 m²

Top and bottom: each 48 m², so 96 m²

Total SA = 96 + 96 = 192 m²

Volume = base area * height = 48 * 3 = 144 m³

✔ Shape ⑨:
Surface Area = 192.0 m²
Volume = 144.0 m³

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Now compile all answers:

① SA: 487.0 ft², Vol: 890.1 ft³
② SA: 254.1 ft², Vol: 261.8 ft³
③ SA: 143.7 yd², Vol: 141.4 yd³
④ SA: 181.5 cm², Vol: 166.4 cm³
⑤ SA: 148.0 cm², Vol: 120.0 cm³
⑥ SA: 207.3 in², Vol: 226.2 in³
⑦ SA: 220.0 m², Vol: 200.0 m³
⑧ SA: 112.0 ft², Vol: 72.0 ft³
⑨ SA: 192.0 m², Vol: 144.0 m³

Final Answer:

① Surface Area: 487.0 ft², Volume: 890.1 ft³
② Surface Area: 254.1 ft², Volume: 261.8 ft³
③ Surface Area: 143.7 yd², Volume: 141.4 yd³
④ Surface Area: 181.5 cm², Volume: 166.4 cm³
⑤ Surface Area: 148.0 cm², Volume: 120.0 cm³
⑥ Surface Area: 207.3 in², Volume: 226.2 in³
⑦ Surface Area: 220.0 m², Volume: 200.0 m³
⑧ Surface Area: 112.0 ft², Volume: 72.0 ft³
⑨ Surface Area: 192.0 m², Volume: 144.0 m³