Surface Area of Composite Shapes

Show your work for each problem.

1.
When two cubes are joined face-to-face, how many faces become hidden?
1. a)6 faces
2. b)2 faces
3. c)4 faces
4. d)1 face
Answer: 2 faces
When two cubes are joined, each contributes one face to the connection, so 2 faces total become hidden (internal).
2.
A cone sits on top of a cylinder (same radius). Which formula should you use for the cone's surface area?
1. a)Just the base: $\pi r^2$
2. b)Total surface area: $\pi r^2 + \pi rl$
3. c)Double the curved surface: $2\pi rl$
4. d)Curved surface area only: $\pi rl$
Answer: Curved surface area only: $\pi rl$
Since the cone sits on the cylinder, its base is hidden. Only use the curved (lateral) surface area: $\pi rl$ .
3.
A cube has a surface area of $150 \text{ cm}^2$ . If a cylindrical hole is drilled through it, removing $20 \text{ cm}^2$ of surface but exposing $50 \text{ cm}^2$ of inner surface, what is the new total surface area?
Answer: 180
$150 - 20 + 50 = 180 \text{ cm}^2$ . When drilling a hole, subtract removed outer surface and add exposed inner surface.
4.
What happens to the total surface area when a hemisphere is placed on top of a cylinder (same radius)?
1. a)Double everything
2. b)Add hemisphere surface area + cylinder surface area
3. c)Just use the hemisphere surface area
4. d)Add hemisphere curved surface, subtract the hidden circle
Answer: Add hemisphere curved surface, subtract the hidden circle
The hemisphere's flat circle and the cylinder's top circle become hidden. Add hemisphere's curved surface ( $2\pi r^2$ ), and subtract the hidden circle ( $\pi r^2$ ) from cylinder's total.
5.
Two cubes, each with side length $5 \text{ cm}$ , are joined face-to-face. What is the total surface area of the composite shape in $\text{cm}^2$ ?
Answer: 250
Total = $2 \times 150 - 2 \times 25 = 300 - 50 = 250 \text{ cm}^2$ . Two cube faces (each $25 \text{ cm}^2$ ) become hidden.
6.
A cylinder (radius $4 \text{ cm}$ , height $10 \text{ cm}$ ) has a cone (same radius, slant height $5 \text{ cm}$ ) on top. Find the total exterior surface area. Use $\pi \approx 3.14$ .
Answer: 364.24
1. Calculate the curved surface area of the cylinder: $2\pi rh = 2 \times 3.14 \times 4 \times 10$ 251.2
2. Calculate the base area of the cylinder: $\pi r^2 = 3.14 \times 4^2$ 50.24
3. Calculate the curved surface area of the cone: $\pi rl = 3.14 \times 4 \times 5$ 62.8
4. Add all surfaces and note: top of cylinder is hidden. Total = CSA cylinder + base + CSA cone 364.24
7.
A cube with side $6 \text{ cm}$ has a cube with side $2 \text{ cm}$ placed on top of it (centered). What is the total surface area in $\text{cm}^2$ ?
Answer: 232
Big cube: $6 \times 6^2 = 216$ . Small cube: $6 \times 2^2 = 24$ . Hidden faces: $2 \times 2^2 = 8$ . Total: $216 + 24 - 8 = 232 \text{ cm}^2$ .
8.
A cube with side $8 \text{ cm}$ has a cylindrical hole (radius $2 \text{ cm}$ ) drilled completely through. Find the total surface area. Use $\pi \approx 3.14$ .
Answer: 459.36
1. Calculate the cube's surface area: $6 \times 8^2$ 384
2. Calculate the area of the two circular holes removed: $2 \times \pi \times 2^2$ 25.12
3. Calculate the curved surface area of the cylindrical hole: $2\pi rh = 2 \times 3.14 \times 2 \times 8$ 100.48
4. Calculate total: Cube SA - removed circles + hole interior 459.36
9.
A rectangular prism ( $10 \times 6 \times 4$ ) has a smaller rectangular prism ( $4 \times 2 \times 2$ ) attached on top. How many square units of surface are hidden?
1. a) $4 \times 2 = 8$ square units
2. b) $2 \times (10 \times 6) = 120$ square units
3. c) $2 \times (4 \times 2) = 16$ square units
4. d) $10 \times 6 = 60$ square units
Answer: $2 \times (4 \times 2) = 16$ square units
Hidden = $2 \times (4 \times 2) = 16$ square units. One face from each prism where they meet becomes internal.
10.
A building has a rectangular base ( $12 \text{ m} \times 8 \text{ m} \times 6 \text{ m}$ ) with a hemispherical dome (radius $4 \text{ m}$ ) on the roof. Calculate the total exterior surface area (excluding the floor). Use $\pi \approx 3.14$ .
Answer: 386.24
1. Calculate the four walls of the building: $2(12 \times 6) + 2(8 \times 6)$ 240
2. Calculate the roof area: $12 \times 8$ 96
3. Calculate the hemisphere curved surface area: $2\pi r^2 = 2 \times 3.14 \times 16$ 100.48
4. Calculate hidden circle (dome base): $\pi r^2 = 3.14 \times 16$ 50.24
5. Total exterior: walls + roof + hemisphere - hidden circle 386.24
11.
A cylinder (radius $5 \text{ cm}$ , height $8 \text{ cm}$ ) has a hemisphere (radius $5 \text{ cm}$ ) on top and a cone (radius $5 \text{ cm}$ , slant height $7 \text{ cm}$ ) on bottom. What is the total surface area? Use $\pi \approx 3.14$ .
Answer: 518.1
Cylinder CSA: $2\pi(5)(8) = 251.2$ . Hemisphere CSA: $2\pi(5)^2 = 157$ . Cone CSA: $\pi(5)(7) = 109.9$ . Total: $251.2 + 157 + 109.9 = 518.1 \text{ cm}^2$ .
12.
A house has walls that form a rectangular prism ( $15 \text{ m} \times 8 \text{ m} \times 5 \text{ m}$ ). The roof is a triangular prism: its triangle has base $8 \text{ m}$ and height $3 \text{ m}$ , and each rectangular roof panel is $15 \text{ m}$ long and $5 \text{ m}$ wide (from the ridge to the eave). Find the total exterior surface area (no floor).
Answer: 404
1. Calculate the four walls: $2(15 \times 5) + 2(8 \times 5)$ 230
2. Calculate the two triangular gable ends: $2 \times \frac{1}{2} \times 8 \times 3$ 24
3. Calculate the two rectangular roof panels: $2 \times (5 \times 15)$ 150
4. Total (walls + gables + roof panels). The top of the rectangular prism is hidden. 404
13.
A cube with side $10 \text{ cm}$ has a cylindrical hole (radius $2 \text{ cm}$ , depth $6 \text{ cm}$ ) drilled into its top face, but not all the way through. The hole has a flat circular bottom. What is the total surface area? Use $\pi \approx 3.14$ .
Answer: 675.36
Cube: $6 \times 10^2 = 600$ . Opening removed: $\pi(2)^2 = 12.56$ . Inner wall: $2\pi(2)(6) = 75.36$ . Bottom of the hole: $\pi(2)^2 = 12.56$ . Total: $600 - 12.56 + 75.36 + 12.56 = 675.36 \text{ cm}^2$ .
14.
Three identical cubes (side $4 \text{ cm}$ ) are joined in a row. What is the total surface area of the composite shape?
1. a) $192 \text{ cm}^2$
2. b) $224 \text{ cm}^2$
3. c) $256 \text{ cm}^2$
4. d) $288 \text{ cm}^2$
Answer: $224 \text{ cm}^2$
Each cube: $6 \times 16 = 96$ . Total: $3 \times 96 = 288$ . Hidden: $4 \times 16 = 64$ . Result: $288 - 64 = 224 \text{ cm}^2$ .
15.
Which statement about composite shape surface area is TRUE?
1. a)Always add all component surface areas together
2. b)Hidden faces must be subtracted from the total
3. c)Components never share faces
4. d)Holes always reduce the total surface area
Answer: Hidden faces must be subtracted from the total
Hidden faces (where shapes connect) must be subtracted. A hole removes two circles but adds its inner wall, which is usually larger, so holes usually ADD surface area. Components DO share faces.