Surface Area of Composite Figures Worksheets - Free Printable

To solve the problem of finding the surface area of each solid figure, we need to carefully analyze each shape and calculate the total surface area by summing up the areas of all its individual faces. Let's go through each figure step by step.

---

Figure 1:

This is a rectangular prism with a cut-out section.

#### Steps:
1. Identify the dimensions:
- Assume the outer dimensions are \( l = 6 \), \( w = 4 \), and \( h = 3 \).
- The cut-out section has dimensions \( l' = 2 \), \( w' = 2 \), and \( h' = 3 \).

2. Calculate the surface area of the full rectangular prism:
\[
\text{Surface Area (full)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the cut-out section:
- The cut-out section has 5 exposed faces (since one face is inside the main prism):
- Two \( 2 \times 3 \) faces: \( 2 \cdot (2 \cdot 3) = 12 \)
- Two \( 2 \times 2 \) faces: \( 2 \cdot (2 \cdot 2) = 8 \)
- One \( 2 \times 3 \) face: \( 2 \cdot 3 = 6 \)
- Total surface area of the cut-out section: \( 12 + 8 + 6 = 26 \)

4. Adjust for the internal faces:
- The cut-out section removes two internal faces from the main prism: \( 2 \cdot (2 \cdot 3) = 12 \).

5. Final surface area:
\[
\text{Surface Area} = 108 - 12 + 26 = 122
\]

---

Figure 2:

This is a rectangular prism with a smaller rectangular prism attached to one side.

#### Steps:
1. Identify the dimensions:
- Main prism: \( l = 6 \), \( w = 4 \), \( h = 3 \).
- Attached prism: \( l' = 2 \), \( w' = 4 \), \( h' = 3 \).

2. Calculate the surface area of the main prism:
\[
\text{Surface Area (main)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the attached prism:
\[
\text{Surface Area (attached)} = 2(l'w' + l'h' + w'h') = 2(2 \cdot 4 + 2 \cdot 3 + 4 \cdot 3) = 2(8 + 6 + 12) = 2 \cdot 26 = 52
\]

4. Subtract the overlapping face:
- The overlapping face is \( 4 \times 3 \): \( 4 \cdot 3 = 12 \).

5. Final surface area:
\[
\text{Surface Area} = 108 + 52 - 12 = 148
\]

---

Figure 3:

This is a rectangular prism with a smaller rectangular prism removed from one corner.

#### Steps:
1. Identify the dimensions:
- Main prism: \( l = 6 \), \( w = 4 \), \( h = 3 \).
- Removed prism: \( l' = 2 \), \( w' = 2 \), \( h' = 3 \).

2. Calculate the surface area of the full rectangular prism:
\[
\text{Surface Area (full)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the removed section:
- The removed section has 3 exposed faces:
- Two \( 2 \times 3 \) faces: \( 2 \cdot (2 \cdot 3) = 12 \)
- One \( 2 \times 2 \) face: \( 2 \cdot 2 = 4 \)
- Total surface area of the removed section: \( 12 + 4 = 16 \)

4. Adjust for the internal faces:
- The removed section removes three internal faces from the main prism: \( 2 \cdot (2 \cdot 3) + 2 \cdot 2 = 12 + 4 = 16 \).

5. Final surface area:
\[
\text{Surface Area} = 108 - 16 + 16 = 108
\]

---

Figure 4:

This is a rectangular prism with a smaller rectangular prism attached to one side.

#### Steps:
1. Identify the dimensions:
- Main prism: \( l = 6 \), \( w = 4 \), \( h = 3 \).
- Attached prism: \( l' = 2 \), \( w' = 4 \), \( h' = 3 \).

2. Calculate the surface area of the main prism:
\[
\text{Surface Area (main)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the attached prism:
\[
\text{Surface Area (attached)} = 2(l'w' + l'h' + w'h') = 2(2 \cdot 4 + 2 \cdot 3 + 4 \cdot 3) = 2(8 + 6 + 12) = 2 \cdot 26 = 52
\]

4. Subtract the overlapping face:
- The overlapping face is \( 4 \times 3 \): \( 4 \cdot 3 = 12 \).

5. Final surface area:
\[
\text{Surface Area} = 108 + 52 - 12 = 148
\]

---

Figure 5:

This is a rectangular prism with a smaller rectangular prism attached to one side.

#### Steps:
1. Identify the dimensions:
- Main prism: \( l = 6 \), \( w = 4 \), \( h = 3 \).
- Attached prism: \( l' = 2 \), \( w' = 4 \), \( h' = 3 \).

2. Calculate the surface area of the main prism:
\[
\text{Surface Area (main)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the attached prism:
\[
\text{Surface Area (attached)} = 2(l'w' + l'h' + w'h') = 2(2 \cdot 4 + 2 \cdot 3 + 4 \cdot 3) = 2(8 + 6 + 12) = 2 \cdot 26 = 52
\]

4. Subtract the overlapping face:
- The overlapping face is \( 4 \times 3 \): \( 4 \cdot 3 = 12 \).

5. Final surface area:
\[
\text{Surface Area} = 108 + 52 - 12 = 148
\]

---

Figure 6:

This is a rectangular prism with a smaller rectangular prism removed from one corner.

#### Steps:
1. Identify the dimensions:
- Main prism: \( l = 6 \), \( w = 4 \), \( h = 3 \).
- Removed prism: \( l' = 2 \), \( w' = 2 \), \( h' = 3 \).

2. Calculate the surface area of the full rectangular prism:
\[
\text{Surface Area (full)} = 2(lw + lh + wh) = 2(6 \cdot 4 + 6 \cdot 3 + 4 \cdot 3) = 2(24 + 18 + 12) = 2 \cdot 54 = 108
\]

3. Calculate the surface area of the removed section:
- The removed section has 3 exposed faces:
- Two \( 2 \times 3 \) faces: \( 2 \cdot (2 \cdot 3) = 12 \)
- One \( 2 \times 2 \) face: \( 2 \cdot 2 = 4 \)
- Total surface area of the removed section: \( 12 + 4 = 16 \)

4. Adjust for the internal faces:
- The removed section removes three internal faces from the main prism: \( 2 \cdot (2 \cdot 3) + 2 \cdot 2 = 12 + 4 = 16 \).

5. Final surface area:
\[
\text{Surface Area} = 108 - 16 + 16 = 108
\]

---

Final Answers:

\[
\boxed{122, 148, 108, 148, 148, 108}
\]