Surface Area of Square Pyramids - Math Practice Worksheet

To solve the surface area of each square pyramid, we will use the formula provided in the example:

\[
\text{Surface Area} = \text{Base Area} + \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height}
\]

Where:
- Base Area = side × side
- Perimeter = 4 × side
- Slant Height is given or can be calculated if needed.

Let's solve each problem step by step.

---

Problem 1:

- Side length (\(s\)) = 8 cm
- Slant height (\(l\)) = 10 cm

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 8 \times 8 = 64 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 8 = 32 \, \text{cm}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 32 \times 10 = 16 \times 10 = 160 \, \text{cm}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 64 + 160 = 224 \, \text{cm}^2
\]

Answer for Problem 1:
\[
\boxed{224 \, \text{cm}^2}
\]

---

Problem 2:

- Side length (\(s\)) = 4 cm
- Slant height (\(l\)) = 9 cm

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 4 \times 4 = 16 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 4 = 16 \, \text{cm}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 16 \times 9 = 8 \times 9 = 72 \, \text{cm}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 16 + 72 = 88 \, \text{cm}^2
\]

Answer for Problem 2:
\[
\boxed{88 \, \text{cm}^2}
\]

---

Problem 3:

- Side length (\(s\)) = 7 cm
- Slant height (\(l\)) = 15 cm

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 7 \times 7 = 49 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 7 = 28 \, \text{cm}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 28 \times 15 = 14 \times 15 = 210 \, \text{cm}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 49 + 210 = 259 \, \text{cm}^2
\]

Answer for Problem 3:
\[
\boxed{259 \, \text{cm}^2}
\]

---

Problem 4:

- Side length (\(s\)) = 3 in
- Slant height (\(l\)) = 11 in

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 3 \times 3 = 9 \, \text{in}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 3 = 12 \, \text{in}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 12 \times 11 = 6 \times 11 = 66 \, \text{in}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 9 + 66 = 75 \, \text{in}^2
\]

Answer for Problem 4:
\[
\boxed{75 \, \text{in}^2}
\]

---

Problem 5:

- Side length (\(s\)) = 6 m
- Slant height (\(l\)) = 14 m

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 6 \times 6 = 36 \, \text{m}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 6 = 24 \, \text{m}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 24 \times 14 = 12 \times 14 = 168 \, \text{m}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 36 + 168 = 204 \, \text{m}^2
\]

Answer for Problem 5:
\[
\boxed{204 \, \text{m}^2}
\]

---

Problem 6:

- Side length (\(s\)) = 9 m
- Slant height (\(l\)) = 12 m

#### Step 1: Calculate the Base Area
\[
\text{Base Area} = s \times s = 9 \times 9 = 81 \, \text{m}^2
\]

#### Step 2: Calculate the Perimeter
\[
\text{Perimeter} = 4 \times s = 4 \times 9 = 36 \, \text{m}
\]

#### Step 3: Calculate the Lateral Surface Area
\[
\text{Lateral Surface Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} = \frac{1}{2} \times 36 \times 12 = 18 \times 12 = 216 \, \text{m}^2
\]

#### Step 4: Calculate the Total Surface Area
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Surface Area} = 81 + 216 = 297 \, \text{m}^2
\]

Answer for Problem 6:
\[
\boxed{297 \, \text{m}^2}
\]

---

Final Answers:

1. \(\boxed{224 \, \text{cm}^2}\)
2. \(\boxed{88 \, \text{cm}^2}\)
3. \(\boxed{259 \, \text{cm}^2}\)
4. \(\boxed{75 \, \text{in}^2}\)
5. \(\boxed{204 \, \text{m}^2}\)
6. \(\boxed{297 \, \text{m}^2}\)