Surface area practice worksheet for prisms and cylinders with diagrams and space for calculations.

Problem Overview:

The task is to calculate the surface area of various 3D shapes, including prisms and cylinders. The shapes are provided in the image, and we need to compute their surface areas using the given dimensions. The value of π is approximated as 3.14.

Key Formulas:

1. Surface Area of a Rectangular Prism:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

2. Surface Area of a Triangular Prism:
\[
\text{Surface Area} = \text{Base Perimeter} \times \text{Height} + 2 \times \text{Base Area}
\]
- Base Area for an equilateral triangle: \( \frac{\sqrt{3}}{4} s^2 \)
- Base Perimeter: \( 3s \)

3. Surface Area of a Cylinder:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi rh
\]
where \( r \) is the radius and \( h \) is the height.

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Step-by-Step Solution:

#### Shape 1: Rectangular Prism
- Dimensions: \( l = 6 \), \( w = 4 \), \( h = 3 \)
- Surface Area:
\[
\text{Surface Area} = 2lw + 2lh + 2wh = 2(6 \times 4) + 2(6 \times 3) + 2(4 \times 3)
\]
\[
= 2(24) + 2(18) + 2(12) = 48 + 36 + 24 = 108
\]

#### Shape 2: Triangular Prism
- Dimensions: Base side \( s = 5 \), Height of prism \( h = 8 \)
- Base Area (Equilateral Triangle):
\[
\text{Base Area} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 5^2 = \frac{\sqrt{3}}{4} \times 25 = \frac{25\sqrt{3}}{4}
\]
Using \( \sqrt{3} \approx 1.732 \):
\[
\text{Base Area} \approx \frac{25 \times 1.732}{4} = \frac{43.3}{4} \approx 10.825
\]
- Base Perimeter:
\[
\text{Base Perimeter} = 3s = 3 \times 5 = 15
\]
- Surface Area:
\[
\text{Surface Area} = \text{Base Perimeter} \times \text{Height} + 2 \times \text{Base Area}
\]
\[
= 15 \times 8 + 2 \times 10.825 = 120 + 21.65 = 141.65
\]

#### Shape 3: Rectangular Prism
- Dimensions: \( l = 10 \), \( w = 5 \), \( h = 4 \)
- Surface Area:
\[
\text{Surface Area} = 2lw + 2lh + 2wh = 2(10 \times 5) + 2(10 \times 4) + 2(5 \times 4)
\]
\[
= 2(50) + 2(40) + 2(20) = 100 + 80 + 40 = 220
\]

#### Shape 4: Triangular Prism
- Dimensions: Base side \( s = 6 \), Height of prism \( h = 9 \)
- Base Area (Equilateral Triangle):
\[
\text{Base Area} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}
\]
Using \( \sqrt{3} \approx 1.732 \):
\[
\text{Base Area} \approx 9 \times 1.732 \approx 15.588
\]
- Base Perimeter:
\[
\text{Base Perimeter} = 3s = 3 \times 6 = 18
\]
- Surface Area:
\[
\text{Surface Area} = \text{Base Perimeter} \times \text{Height} + 2 \times \text{Base Area}
\]
\[
= 18 \times 9 + 2 \times 15.588 = 162 + 31.176 \approx 193.176
\]

#### Shape 5: Cylinder
- Dimensions: Radius \( r = 3 \), Height \( h = 7 \)
- Surface Area:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi rh
\]
\[
= 2\pi (3^2) + 2\pi (3)(7) = 2\pi (9) + 2\pi (21) = 18\pi + 42\pi = 60\pi
\]
Using \( \pi \approx 3.14 \):
\[
\text{Surface Area} \approx 60 \times 3.14 = 188.4
\]

#### Shape 6: Rectangular Prism
- Dimensions: \( l = 8 \), \( w = 6 \), \( h = 5 \)
- Surface Area:
\[
\text{Surface Area} = 2lw + 2lh + 2wh = 2(8 \times 6) + 2(8 \times 5) + 2(6 \times 5)
\]
\[
= 2(48) + 2(40) + 2(30) = 96 + 80 + 60 = 236
\]

#### Shape 7: Cylinder
- Dimensions: Radius \( r = 4 \), Height \( h = 10 \)
- Surface Area:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi rh
\]
\[
= 2\pi (4^2) + 2\pi (4)(10) = 2\pi (16) + 2\pi (40) = 32\pi + 80\pi = 112\pi
\]
Using \( \pi \approx 3.14 \):
\[
\text{Surface Area} \approx 112 \times 3.14 = 351.68
\]

#### Shape 8: Triangular Prism
- Dimensions: Base side \( s = 7 \), Height of prism \( h = 12 \)
- Base Area (Equilateral Triangle):
\[
\text{Base Area} = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 7^2 = \frac{\sqrt{3}}{4} \times 49 = \frac{49\sqrt{3}}{4}
\]
Using \( \sqrt{3} \approx 1.732 \):
\[
\text{Base Area} \approx \frac{49 \times 1.732}{4} = \frac{84.868}{4} \approx 21.217
\]
- Base Perimeter:
\[
\text{Base Perimeter} = 3s = 3 \times 7 = 21
\]
- Surface Area:
\[
\text{Surface Area} = \text{Base Perimeter} \times \text{Height} + 2 \times \text{Base Area}
\]
\[
= 21 \times 12 + 2 \times 21.217 = 252 + 42.434 \approx 294.434
\]

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Final Answers:

1. \( \boxed{108} \)
2. \( \boxed{141.65} \)
3. \( \boxed{220} \)
4. \( \boxed{193.176} \)
5. \( \boxed{188.4} \)
6. \( \boxed{236} \)
7. \( \boxed{351.68} \)
8. \( \boxed{294.434} \)