Worksheet for calculating the surface area of different geometric solids like prisms and cylinders.

To solve the problem of finding the surface area of the given solid shapes, we will calculate the surface area for each shape step by step. Let's go through each one:

---

1) A square-based prism

- Dimensions: Base side = 9 cm, Height = 14 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(9 \times 9 = 81 \, \text{cm}^2\)
- Perimeter of Base = \(4 \times 9 = 36 \, \text{cm}\)
- Surface Area = \(2 \times 81 + 36 \times 14 = 162 + 504 = 666 \, \text{cm}^2\)

Answer: \(666 \, \text{cm}^2\)

---

2) A rectangular prism

- Dimensions: Length = 12 cm, Width = 2 cm, Height = 8 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (lw + lh + wh)
\]
- \(lw = 12 \times 2 = 24\)
- \(lh = 12 \times 8 = 96\)
- \(wh = 2 \times 8 = 16\)
- Surface Area = \(2 \times (24 + 96 + 16) = 2 \times 136 = 272 \, \text{cm}^2\)

Answer: \(272 \, \text{cm}^2\)

---

3) A triangular prism

- Base: Right triangle with legs 5 cm and 4 cm, hypotenuse 11 cm
- Height of Prism: 11 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(\frac{1}{2} \times 5 \times 4 = 10 \, \text{cm}^2\)
- Perimeter of Base = \(5 + 4 + 11 = 20 \, \text{cm}\)
- Surface Area = \(2 \times 10 + 20 \times 11 = 20 + 220 = 240 \, \text{cm}^2\)

Answer: \(240 \, \text{cm}^2\)

---

4) A triangular prism

- Base: Triangle with sides 6 cm, 8 cm, and 10 cm (right triangle)
- Height of Prism: Not given, assume it is the same as one of the sides for simplicity (e.g., 8 cm)
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(\frac{1}{2} \times 6 \times 8 = 24 \, \text{cm}^2\)
- Perimeter of Base = \(6 + 8 + 10 = 24 \, \text{cm}\)
- Surface Area = \(2 \times 24 + 24 \times 8 = 48 + 192 = 240 \, \text{cm}^2\)

Answer: \(240 \, \text{cm}^2\)

---

5) A right parallelepiped

- Dimensions: Length = 11 cm, Width = 8 cm, Height = 7 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (lw + lh + wh)
\]
- \(lw = 11 \times 8 = 88\)
- \(lh = 11 \times 7 = 77\)
- \(wh = 8 \times 7 = 56\)
- Surface Area = \(2 \times (88 + 77 + 56) = 2 \times 221 = 442 \, \text{cm}^2\)

Answer: \(442 \, \text{cm}^2\)

---

6) A triangular prism

- Base: Triangle with sides 11 cm, 8 cm, and 5 cm
- Height of Prism: 15 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(\sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \frac{11+8+5}{2} = 12\)
\[
\text{Base Area} = \sqrt{12(12-11)(12-8)(12-5)} = \sqrt{12 \times 1 \times 4 \times 7} = \sqrt{336} \approx 18.33 \, \text{cm}^2
\]
- Perimeter of Base = \(11 + 8 + 5 = 24 \, \text{cm}\)
- Surface Area = \(2 \times 18.33 + 24 \times 15 = 36.66 + 360 = 396.66 \, \text{cm}^2\)

Answer: \(396.66 \, \text{cm}^2\)

---

7) A right trapezium-based prism

- Base: Trapezium with parallel sides 24 cm and 10 cm, height 14 cm
- Height of Prism: 18 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(\frac{1}{2} \times (24 + 10) \times 14 = \frac{1}{2} \times 34 \times 14 = 238 \, \text{cm}^2\)
- Perimeter of Base = \(24 + 10 + 14 + \sqrt{(24-10)^2 + 14^2} = 24 + 10 + 14 + \sqrt{14^2 + 14^2} = 48 + 14\sqrt{2} \approx 69.6 \, \text{cm}\)
- Surface Area = \(2 \times 238 + 69.6 \times 18 \approx 476 + 1252.8 = 1728.8 \, \text{cm}^2\)

Answer: \(1728.8 \, \text{cm}^2\)

---

8) A trapezium-based prism

- Base: Trapezium with parallel sides 7 cm and 4 cm, height 4 cm
- Height of Prism: Not given, assume it is the same as one of the sides for simplicity (e.g., 4 cm)
- Surface Area Formula:
\[
\text{Surface Area} = 2 \times (\text{Base Area}) + \text{Perimeter of Base} \times \text{Height}
\]
- Base Area = \(\frac{1}{2} \times (7 + 4) \times 4 = \frac{1}{2} \times 11 \times 4 = 22 \, \text{cm}^2\)
- Perimeter of Base = \(7 + 4 + 4 + \sqrt{(7-4)^2 + 4^2} = 15 + \sqrt{9 + 16} = 15 + 5 = 20 \, \text{cm}\)
- Surface Area = \(2 \times 22 + 20 \times 4 = 44 + 80 = 124 \, \text{cm}^2\)

Answer: \(124 \, \text{cm}^2\)

---

9) A cylinder

- Radius: 3 cm, Height: 6 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
- Surface Area = \(2\pi \times 3 \times (6 + 3) = 2\pi \times 3 \times 9 = 54\pi \approx 169.65 \, \text{cm}^2\)

Answer: \(169.65 \, \text{cm}^2\)

---

10) A cylinder

- Radius: 10 cm, Height: 20 cm
- Surface Area Formula:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
- Surface Area = \(2\pi \times 10 \times (20 + 10) = 2\pi \times 10 \times 30 = 600\pi \approx 1884.96 \, \text{cm}^2\)

Answer: \(1884.96 \, \text{cm}^2\)

---

11) A half cylinder

- Radius: 5.5 cm, Height: 11 cm
- Surface Area Formula:
\[
\text{Surface Area} = \pi r(h + r) + 2rh
\]
- Surface Area = \(\pi \times 5.5 \times (11 + 5.5) + 2 \times 5.5 \times 11 = \pi \times 5.5 \times 16.5 + 121 \approx 288.98 + 121 = 409.98 \, \text{cm}^2\)

Answer: \(409.98 \, \text{cm}^2\)

---

12) A three-quarter cylinder

- Radius: 4.5 cm, Height: 9 cm
- Surface Area Formula:
\[
\text{Surface Area} = \frac{3}{4} \times (2\pi r(h + r)) + 2rh
\]
- Surface Area = \(\frac{3}{4} \times (2\pi \times 4.5 \times (9 + 4.5)) + 2 \times 4.5 \times 9 = \frac{3}{4} \times (2\pi \times 4.5 \times 13.5) + 81 \approx \frac{3}{4} \times 381.70 + 81 \approx 286.28 + 81 = 367.28 \, \text{cm}^2\)

Answer: \(367.28 \, \text{cm}^2\)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & 666 \, \text{cm}^2 \\
2) & 272 \, \text{cm}^2 \\
3) & 240 \, \text{cm}^2 \\
4) & 240 \, \text{cm}^2 \\
5) & 442 \, \text{cm}^2 \\
6) & 396.66 \, \text{cm}^2 \\
7) & 1728.8 \, \text{cm}^2 \\
8) & 124 \, \text{cm}^2 \\
9) & 169.65 \, \text{cm}^2 \\
10) & 1884.96 \, \text{cm}^2 \\
11) & 409.98 \, \text{cm}^2 \\
12) & 367.28 \, \text{cm}^2 \\
\end{array}
}
\]