Surface area calculation worksheet for prisms and cylinders with diagrams and space for answers.

To solve the problem of finding the surface area of each shape in the image, we need to calculate the surface area for prisms and cylinders. Below is a step-by-step explanation for each shape:

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1. Cylinder

The formula for the surface area of a cylinder is:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi rh
\]
where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.

#### Given:
- Radius (\( r \)) = 3 cm,
- Height (\( h \)) = 5 cm.

#### Calculation:
\[
\text{Surface Area} = 2\pi (3)^2 + 2\pi (3)(5)
\]
\[
= 2\pi (9) + 2\pi (15)
\]
\[
= 18\pi + 30\pi
\]
\[
= 48\pi \, \text{cm}^2
\]

#### Answer:
\[
\boxed{48\pi}
\]

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2. Rectangular Prism

The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.

#### Given:
- Length (\( l \)) = 6 cm,
- Width (\( w \)) = 4 cm,
- Height (\( h \)) = 3 cm.

#### Calculation:
\[
\text{Surface Area} = 2(6)(4) + 2(6)(3) + 2(4)(3)
\]
\[
= 2(24) + 2(18) + 2(12)
\]
\[
= 48 + 36 + 24
\]
\[
= 108 \, \text{cm}^2
\]

#### Answer:
\[
\boxed{108}
\]

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3. Triangular Prism

The formula for the surface area of a triangular prism is:
\[
\text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Height of Prism}
\]
For a triangular prism with a right triangle as the base:
- The base area is given by \( \frac{1}{2} \times \text{base} \times \text{height} \),
- The perimeter of the base is the sum of all sides of the triangle.

#### Given:
- Base of triangle (\( b \)) = 4 cm,
- Height of triangle (\( h_{\text{triangle}} \)) = 3 cm,
- Slant height of triangle (\( s \)) = 5 cm (hypotenuse),
- Height of prism (\( H \)) = 7 cm.

#### Step 1: Calculate the base area.
\[
\text{Base Area} = \frac{1}{2} \times 4 \times 3 = 6 \, \text{cm}^2
\]

#### Step 2: Calculate the perimeter of the base.
\[
\text{Perimeter} = 4 + 3 + 5 = 12 \, \text{cm}
\]

#### Step 3: Calculate the surface area.
\[
\text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter} \times \text{Height of Prism}
\]
\[
= 2 \times 6 + 12 \times 7
\]
\[
= 12 + 84
\]
\[
= 96 \, \text{cm}^2
\]

#### Answer:
\[
\boxed{96}
\]

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4. Rectangular Prism

The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]

#### Given:
- Length (\( l \)) = 8 cm,
- Width (\( w \)) = 5 cm,
- Height (\( h \)) = 3 cm.

#### Calculation:
\[
\text{Surface Area} = 2(8)(5) + 2(8)(3) + 2(5)(3)
\]
\[
= 2(40) + 2(24) + 2(15)
\]
\[
= 80 + 48 + 30
\]
\[
= 158 \, \text{cm}^2
\]

#### Answer:
\[
\boxed{158}
\]

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5. Triangular Prism

The formula for the surface area of a triangular prism is:
\[
\text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Height of Prism}
\]

#### Given:
- Base of triangle (\( b \)) = 6 cm,
- Height of triangle (\( h_{\text{triangle}} \)) = 4 cm,
- Slant height of triangle (\( s \)) = 5 cm (hypotenuse),
- Height of prism (\( H \)) = 10 cm.

#### Step 1: Calculate the base area.
\[
\text{Base Area} = \frac{1}{2} \times 6 \times 4 = 12 \, \text{cm}^2
\]

#### Step 2: Calculate the perimeter of the base.
\[
\text{Perimeter} = 6 + 4 + 5 = 15 \, \text{cm}
\]

#### Step 3: Calculate the surface area.
\[
\text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter} \times \text{Height of Prism}
\]
\[
= 2 \times 12 + 15 \times 10
\]
\[
= 24 + 150
\]
\[
= 174 \, \text{cm}^2
\]

#### Answer:
\[
\boxed{174}
\]

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6. Cylinder

The formula for the surface area of a cylinder is:
\[
\text{Surface Area} = 2\pi r^2 + 2\pi rh
\]

#### Given:
- Radius (\( r \)) = 2 cm,
- Height (\( h \)) = 6 cm.

#### Calculation:
\[
\text{Surface Area} = 2\pi (2)^2 + 2\pi (2)(6)
\]
\[
= 2\pi (4) + 2\pi (12)
\]
\[
= 8\pi + 24\pi
\]
\[
= 32\pi \, \text{cm}^2
\]

#### Answer:
\[
\boxed{32\pi}
\]

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Final Answers:

1. \( \boxed{48\pi} \)
2. \( \boxed{108} \)
3. \( \boxed{96} \)
4. \( \boxed{158} \)
5. \( \boxed{174} \)
6. \( \boxed{32\pi} \)

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These are the detailed solutions for each shape. If you have any further questions, feel free to ask!