Volume, Surface Area Rectangular Prisms Worksheet - Free Printable

To solve the problem of finding the surface area for each shape, we need to use the appropriate formulas for prisms and cylinders. Let's go through each shape step by step.

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1. Rectangular Prism

Dimensions:
- Length (\( l \)) = 6 ft
- Width (\( w \)) = 4 ft
- Height (\( h \)) = 12 ft

Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]

Calculation:
\[ \text{Surface Area} = 2(6 \cdot 4 + 6 \cdot 12 + 4 \cdot 12) \]
\[ = 2(24 + 72 + 48) \]
\[ = 2(144) \]
\[ = 288 \, \text{ft}^2 \]

Answer:
\[ \boxed{288 \, \text{ft}^2} \]

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2. Triangular Prism

Dimensions:
- Base (\( b \)) = 6 yd
- Height of triangle (\( h_{\text{triangle}} \)) = 5 yd
- Slant height (\( s \)) = 7 yd
- Prism height (\( H \)) = 3 yd

Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Height of Prism} \]

Step 1: Calculate the Base Area of the Triangle:
\[ \text{Base Area} = \frac{1}{2} \times b \times h_{\text{triangle}} \]
\[ = \frac{1}{2} \times 6 \times 5 \]
\[ = 15 \, \text{yd}^2 \]

Step 2: Calculate the Perimeter of the Base Triangle:
The base is an isosceles triangle with sides \( b = 6 \, \text{yd} \), \( s = 7 \, \text{yd} \), and \( s = 7 \, \text{yd} \).
\[ \text{Perimeter} = 6 + 7 + 7 = 20 \, \text{yd} \]

Step 3: Calculate the Lateral Surface Area:
\[ \text{Lateral Surface Area} = \text{Perimeter of Base} \times \text{Height of Prism} \]
\[ = 20 \times 3 \]
\[ = 60 \, \text{yd}^2 \]

Step 4: Total Surface Area:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 15 + 60 \]
\[ = 30 + 60 \]
\[ = 90 \, \text{yd}^2 \]

Answer:
\[ \boxed{90 \, \text{yd}^2} \]

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3. Rectangular Prism

Dimensions:
- Length (\( l \)) = 8 in
- Width (\( w \)) = 6 in
- Height (\( h \)) = 13 in

Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]

Calculation:
\[ \text{Surface Area} = 2(8 \cdot 6 + 8 \cdot 13 + 6 \cdot 13) \]
\[ = 2(48 + 104 + 78) \]
\[ = 2(230) \]
\[ = 460 \, \text{in}^2 \]

Answer:
\[ \boxed{460 \, \text{in}^2} \]

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4. Trapezoidal Prism

Dimensions:
- Bases of trapezoid (\( b_1 \)) = 12 in, (\( b_2 \)) = 9 in
- Height of trapezoid (\( h_{\text{trapezoid}} \)) = 8 in
- Slant height (\( s \)) = 15 in
- Prism height (\( H \)) = 6 in

Formula for Surface Area of a Trapezoidal Prism:
\[ \text{Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Perimeter of Trapezoid} \times \text{Height of Prism} \]

Step 1: Calculate the Area of the Trapezoid:
\[ \text{Area of Trapezoid} = \frac{1}{2} \times (b_1 + b_2) \times h_{\text{trapezoid}} \]
\[ = \frac{1}{2} \times (12 + 9) \times 8 \]
\[ = \frac{1}{2} \times 21 \times 8 \]
\[ = 84 \, \text{in}^2 \]

Step 2: Calculate the Perimeter of the Trapezoid:
The perimeter includes the two bases and the two slant heights.
\[ \text{Perimeter} = b_1 + b_2 + 2s \]
\[ = 12 + 9 + 2 \times 15 \]
\[ = 12 + 9 + 30 \]
\[ = 51 \, \text{in} \]

Step 3: Calculate the Lateral Surface Area:
\[ \text{Lateral Surface Area} = \text{Perimeter of Trapezoid} \times \text{Height of Prism} \]
\[ = 51 \times 6 \]
\[ = 306 \, \text{in}^2 \]

Step 4: Total Surface Area:
\[ \text{Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Lateral Surface Area} \]
\[ = 2 \times 84 + 306 \]
\[ = 168 + 306 \]
\[ = 474 \, \text{in}^2 \]

Answer:
\[ \boxed{474 \, \text{in}^2} \]

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5. Rectangular Prism

Dimensions:
- Length (\( l \)) = 9 ft
- Width (\( w \)) = 5 ft
- Height (\( h \)) = 36 ft

Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]

Calculation:
\[ \text{Surface Area} = 2(9 \cdot 5 + 9 \cdot 36 + 5 \cdot 36) \]
\[ = 2(45 + 324 + 180) \]
\[ = 2(549) \]
\[ = 1098 \, \text{ft}^2 \]

Answer:
\[ \boxed{1098 \, \text{ft}^2} \]

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6. Cylinder

Dimensions:
- Radius (\( r \)) = 10 m
- Height (\( h \)) = 14 m

Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]

Step 1: Calculate the Area of the Two Bases:
\[ \text{Area of Bases} = 2\pi r^2 \]
\[ = 2\pi (10)^2 \]
\[ = 2\pi \times 100 \]
\[ = 200\pi \, \text{m}^2 \]

Step 2: Calculate the Lateral Surface Area:
\[ \text{Lateral Surface Area} = 2\pi rh \]
\[ = 2\pi (10)(14) \]
\[ = 2\pi \times 140 \]
\[ = 280\pi \, \text{m}^2 \]

Step 3: Total Surface Area:
\[ \text{Surface Area} = 200\pi + 280\pi \]
\[ = 480\pi \, \text{m}^2 \]

Answer:
\[ \boxed{480\pi \, \text{m}^2} \]

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7. Cylinder

Dimensions:
- Radius (\( r \)) = 3 cm
- Height (\( h \)) = 11 cm

Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]

Step 1: Calculate the Area of the Two Bases:
\[ \text{Area of Bases} = 2\pi r^2 \]
\[ = 2\pi (3)^2 \]
\[ = 2\pi \times 9 \]
\[ = 18\pi \, \text{cm}^2 \]

Step 2: Calculate the Lateral Surface Area:
\[ \text{Lateral Surface Area} = 2\pi rh \]
\[ = 2\pi (3)(11) \]
\[ = 2\pi \times 33 \]
\[ = 66\pi \, \text{cm}^2 \]

Step 3: Total Surface Area:
\[ \text{Surface Area} = 18\pi + 66\pi \]
\[ = 84\pi \, \text{cm}^2 \]

Answer:
\[ \boxed{84\pi \, \text{cm}^2} \]

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8. Rectangular Prism

Dimensions:
- Length (\( l \)) = 7 ft
- Width (\( w \)) = 4 ft
- Height (\( h \)) = 14 ft

Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]

Calculation:
\[ \text{Surface Area} = 2(7 \cdot 4 + 7 \cdot 14 + 4 \cdot 14) \]
\[ = 2(28 + 98 + 56) \]
\[ = 2(182) \]
\[ = 364 \, \text{ft}^2 \]

Answer:
\[ \boxed{364 \, \text{ft}^2} \]

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9. Triangular Prism

Dimensions:
- Base (\( b \)) = 10 ft
- Height of triangle (\( h_{\text{triangle}} \)) = 12 ft
- Slant height (\( s \)) = 13 ft
- Prism height (\( H \)) = 17 ft

Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Height of Prism} \]

Step 1: Calculate the Base Area of the Triangle:
\[ \text{Base Area} = \frac{1}{2} \times b \times h_{\text{triangle}} \]
\[ = \frac{1}{2} \times 10 \times 12 \]
\[ = 60 \, \text{ft}^2 \]

Step 2: Calculate the Perimeter of the Base Triangle:
The base is a right triangle with sides \( b = 10 \, \text{ft} \), \( h_{\text{triangle}} = 12 \, \text{ft} \), and hypotenuse \( s = 13 \, \text{ft} \).
\[ \text{Perimeter} = 10 + 12 + 13 = 35 \, \text{ft} \]

Step 3: Calculate the Lateral Surface Area:
\[ \text{Lateral Surface Area} = \text{Perimeter of Base} \times \text{Height of Prism} \]
\[ = 35 \times 17 \]
\[ = 595 \, \text{ft}^2 \]

Step 4: Total Surface Area:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 60 + 595 \]
\[ = 120 + 595 \]
\[ = 715 \, \text{ft}^2 \]

Answer:
\[ \boxed{715 \, \text{ft}^2} \]

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Final Answers:

1. \( \boxed{288 \, \text{ft}^2} \)
2. \( \boxed{90 \, \text{yd}^2} \)
3. \( \boxed{460 \, \text{in}^2} \)
4. \( \boxed{474 \, \text{in}^2} \)
5. \( \boxed{1098 \, \text{ft}^2} \)
6. \( \boxed{480\pi \, \text{m}^2} \)
7. \( \boxed{84\pi \, \text{cm}^2} \)
8. \( \boxed{364 \, \text{ft}^2} \)
9. \( \boxed{715 \, \text{ft}^2} \)