Printable math worksheet for calculating surface area of prisms and cylinders.
Educational worksheet: Volume and surface area of rectangular prisms - Math Worksheets. Download and print for classroom or home learning activities.
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Step-by-step solution for: Volume and surface area of rectangular prisms - Math Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Volume and surface area of rectangular prisms - Math Worksheets
To solve the problem of finding the surface area for each shape in the worksheet, we need to use the appropriate formulas for prisms and cylinders. Let's go through each shape step by step.
---
Dimensions:
- Length = 6 ft
- Width = 4 ft
- Height = 12 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(6 \cdot 4 + 6 \cdot 12 + 4 \cdot 12) \]
\[ = 2(24 + 72 + 48) \]
\[ = 2(144) \]
\[ = 288 \, \text{ft}^2 \]
Answer:
\[ \boxed{288 \, \text{ft}^2} \]
---
Dimensions:
- Base of triangle = 6 yd
- Height of triangle = 5 yd
- Slant height of triangle = 7 yd
- Length of prism = 10 yd
Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Length} \]
Step 1: Calculate the area of one triangular base:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
\[ = \frac{1}{2} \times 6 \times 5 \]
\[ = 15 \, \text{yd}^2 \]
Step 2: Calculate the perimeter of the triangular base:
The sides of the triangle are 6 yd, 5 yd, and 7 yd.
\[ \text{Perimeter} = 6 + 5 + 7 = 18 \, \text{yd} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 18 \times 10 \]
\[ = 180 \, \text{yd}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 15 + 180 \]
\[ = 30 + 180 \]
\[ = 210 \, \text{yd}^2 \]
Answer:
\[ \boxed{210 \, \text{yd}^2} \]
---
Dimensions:
- Length = 8 m
- Width = 6 m
- Height = 13 m
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(8 \cdot 6 + 8 \cdot 13 + 6 \cdot 13) \]
\[ = 2(48 + 104 + 78) \]
\[ = 2(230) \]
\[ = 460 \, \text{m}^2 \]
Answer:
\[ \boxed{460 \, \text{m}^2} \]
---
Dimensions:
- Bases of trapezoid = 9 m and 12 m
- Height of trapezoid = 8 m
- Slant height of trapezoid = 15 m
- Length of prism = 15 m
Formula for Surface Area of a Trapezoidal Prism:
\[ \text{Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Perimeter of Trapezoid} \times \text{Length} \]
Step 1: Calculate the area of one trapezoidal base:
\[ \text{Area of Trapezoid} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} \]
\[ = \frac{1}{2} \times (9 + 12) \times 8 \]
\[ = \frac{1}{2} \times 21 \times 8 \]
\[ = 84 \, \text{m}^2 \]
Step 2: Calculate the perimeter of the trapezoid:
The sides of the trapezoid are 9 m, 12 m, and two slant heights of 15 m each.
\[ \text{Perimeter} = 9 + 12 + 15 + 15 = 51 \, \text{m} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 51 \times 15 \]
\[ = 765 \, \text{m}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Lateral Surface Area} \]
\[ = 2 \times 84 + 765 \]
\[ = 168 + 765 \]
\[ = 933 \, \text{m}^2 \]
Answer:
\[ \boxed{933 \, \text{m}^2} \]
---
Dimensions:
- Length = 9 ft
- Width = 9 ft
- Height = 36 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(9 \cdot 9 + 9 \cdot 36 + 9 \cdot 36) \]
\[ = 2(81 + 324 + 324) \]
\[ = 2(729) \]
\[ = 1458 \, \text{ft}^2 \]
Answer:
\[ \boxed{1458 \, \text{ft}^2} \]
---
Dimensions:
- Radius = 10 m
- Height = 10 m
Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]
Substitute the values:
\[ \text{Surface Area} = 2\pi (10)^2 + 2\pi (10)(10) \]
\[ = 2\pi (100) + 2\pi (100) \]
\[ = 200\pi + 200\pi \]
\[ = 400\pi \, \text{m}^2 \]
Answer:
\[ \boxed{400\pi \, \text{m}^2} \]
---
Dimensions:
- Radius = 3 cm
- Height = 11 cm
Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]
Substitute the values:
\[ \text{Surface Area} = 2\pi (3)^2 + 2\pi (3)(11) \]
\[ = 2\pi (9) + 2\pi (33) \]
\[ = 18\pi + 66\pi \]
\[ = 84\pi \, \text{cm}^2 \]
Answer:
\[ \boxed{84\pi \, \text{cm}^2} \]
---
Dimensions:
- Length = 7 ft
- Width = 7 ft
- Height = 14 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(7 \cdot 7 + 7 \cdot 14 + 7 \cdot 14) \]
\[ = 2(49 + 98 + 98) \]
\[ = 2(245) \]
\[ = 490 \, \text{ft}^2 \]
Answer:
\[ \boxed{490 \, \text{ft}^2} \]
---
Dimensions:
- Base of triangle = 10 ft
- Height of triangle = 12 ft
- Slant height of triangle = 13 ft
- Length of prism = 15 ft
Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter of Base} \times \text{Length} \]
Step 1: Calculate the area of one triangular base:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
\[ = \frac{1}{2} \times 10 \times 12 \]
\[ = 60 \, \text{ft}^2 \]
Step 2: Calculate the perimeter of the triangular base:
The sides of the triangle are 10 ft, 12 ft, and 13 ft.
\[ \text{Perimeter} = 10 + 12 + 13 = 35 \, \text{ft} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 35 \times 15 \]
\[ = 525 \, \text{ft}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 60 + 525 \]
\[ = 120 + 525 \]
\[ = 645 \, \text{ft}^2 \]
Answer:
\[ \boxed{645 \, \text{ft}^2} \]
---
1. \( \boxed{288 \, \text{ft}^2} \)
2. \( \boxed{210 \, \text{yd}^2} \)
3. \( \boxed{460 \, \text{m}^2} \)
4. \( \boxed{933 \, \text{m}^2} \)
5. \( \boxed{1458 \, \text{ft}^2} \)
6. \( \boxed{400\pi \, \text{m}^2} \)
7. \( \boxed{84\pi \, \text{cm}^2} \)
8. \( \boxed{490 \, \text{ft}^2} \)
9. \( \boxed{645 \, \text{ft}^2} \)
---
1. Rectangular Prism
Dimensions:
- Length = 6 ft
- Width = 4 ft
- Height = 12 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(6 \cdot 4 + 6 \cdot 12 + 4 \cdot 12) \]
\[ = 2(24 + 72 + 48) \]
\[ = 2(144) \]
\[ = 288 \, \text{ft}^2 \]
Answer:
\[ \boxed{288 \, \text{ft}^2} \]
---
2. Triangular Prism
Dimensions:
- Base of triangle = 6 yd
- Height of triangle = 5 yd
- Slant height of triangle = 7 yd
- Length of prism = 10 yd
Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Length} \]
Step 1: Calculate the area of one triangular base:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
\[ = \frac{1}{2} \times 6 \times 5 \]
\[ = 15 \, \text{yd}^2 \]
Step 2: Calculate the perimeter of the triangular base:
The sides of the triangle are 6 yd, 5 yd, and 7 yd.
\[ \text{Perimeter} = 6 + 5 + 7 = 18 \, \text{yd} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 18 \times 10 \]
\[ = 180 \, \text{yd}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 15 + 180 \]
\[ = 30 + 180 \]
\[ = 210 \, \text{yd}^2 \]
Answer:
\[ \boxed{210 \, \text{yd}^2} \]
---
3. Rectangular Prism
Dimensions:
- Length = 8 m
- Width = 6 m
- Height = 13 m
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(8 \cdot 6 + 8 \cdot 13 + 6 \cdot 13) \]
\[ = 2(48 + 104 + 78) \]
\[ = 2(230) \]
\[ = 460 \, \text{m}^2 \]
Answer:
\[ \boxed{460 \, \text{m}^2} \]
---
4. Trapezoidal Prism
Dimensions:
- Bases of trapezoid = 9 m and 12 m
- Height of trapezoid = 8 m
- Slant height of trapezoid = 15 m
- Length of prism = 15 m
Formula for Surface Area of a Trapezoidal Prism:
\[ \text{Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Perimeter of Trapezoid} \times \text{Length} \]
Step 1: Calculate the area of one trapezoidal base:
\[ \text{Area of Trapezoid} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} \]
\[ = \frac{1}{2} \times (9 + 12) \times 8 \]
\[ = \frac{1}{2} \times 21 \times 8 \]
\[ = 84 \, \text{m}^2 \]
Step 2: Calculate the perimeter of the trapezoid:
The sides of the trapezoid are 9 m, 12 m, and two slant heights of 15 m each.
\[ \text{Perimeter} = 9 + 12 + 15 + 15 = 51 \, \text{m} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 51 \times 15 \]
\[ = 765 \, \text{m}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Area of Trapezoid} + \text{Lateral Surface Area} \]
\[ = 2 \times 84 + 765 \]
\[ = 168 + 765 \]
\[ = 933 \, \text{m}^2 \]
Answer:
\[ \boxed{933 \, \text{m}^2} \]
---
5. Rectangular Prism
Dimensions:
- Length = 9 ft
- Width = 9 ft
- Height = 36 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(9 \cdot 9 + 9 \cdot 36 + 9 \cdot 36) \]
\[ = 2(81 + 324 + 324) \]
\[ = 2(729) \]
\[ = 1458 \, \text{ft}^2 \]
Answer:
\[ \boxed{1458 \, \text{ft}^2} \]
---
6. Cylinder
Dimensions:
- Radius = 10 m
- Height = 10 m
Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]
Substitute the values:
\[ \text{Surface Area} = 2\pi (10)^2 + 2\pi (10)(10) \]
\[ = 2\pi (100) + 2\pi (100) \]
\[ = 200\pi + 200\pi \]
\[ = 400\pi \, \text{m}^2 \]
Answer:
\[ \boxed{400\pi \, \text{m}^2} \]
---
7. Cylinder
Dimensions:
- Radius = 3 cm
- Height = 11 cm
Formula for Surface Area of a Cylinder:
\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]
Substitute the values:
\[ \text{Surface Area} = 2\pi (3)^2 + 2\pi (3)(11) \]
\[ = 2\pi (9) + 2\pi (33) \]
\[ = 18\pi + 66\pi \]
\[ = 84\pi \, \text{cm}^2 \]
Answer:
\[ \boxed{84\pi \, \text{cm}^2} \]
---
8. Rectangular Prism
Dimensions:
- Length = 7 ft
- Width = 7 ft
- Height = 14 ft
Formula for Surface Area of a Rectangular Prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
Substitute the values:
\[ \text{Surface Area} = 2(7 \cdot 7 + 7 \cdot 14 + 7 \cdot 14) \]
\[ = 2(49 + 98 + 98) \]
\[ = 2(245) \]
\[ = 490 \, \text{ft}^2 \]
Answer:
\[ \boxed{490 \, \text{ft}^2} \]
---
9. Triangular Prism
Dimensions:
- Base of triangle = 10 ft
- Height of triangle = 12 ft
- Slant height of triangle = 13 ft
- Length of prism = 15 ft
Formula for Surface Area of a Triangular Prism:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter of Base} \times \text{Length} \]
Step 1: Calculate the area of one triangular base:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
\[ = \frac{1}{2} \times 10 \times 12 \]
\[ = 60 \, \text{ft}^2 \]
Step 2: Calculate the perimeter of the triangular base:
The sides of the triangle are 10 ft, 12 ft, and 13 ft.
\[ \text{Perimeter} = 10 + 12 + 13 = 35 \, \text{ft} \]
Step 3: Calculate the lateral surface area:
\[ \text{Lateral Surface Area} = \text{Perimeter} \times \text{Length} \]
\[ = 35 \times 15 \]
\[ = 525 \, \text{ft}^2 \]
Step 4: Calculate the total surface area:
\[ \text{Total Surface Area} = 2 \times \text{Base Area} + \text{Lateral Surface Area} \]
\[ = 2 \times 60 + 525 \]
\[ = 120 + 525 \]
\[ = 645 \, \text{ft}^2 \]
Answer:
\[ \boxed{645 \, \text{ft}^2} \]
---
Final Answers:
1. \( \boxed{288 \, \text{ft}^2} \)
2. \( \boxed{210 \, \text{yd}^2} \)
3. \( \boxed{460 \, \text{m}^2} \)
4. \( \boxed{933 \, \text{m}^2} \)
5. \( \boxed{1458 \, \text{ft}^2} \)
6. \( \boxed{400\pi \, \text{m}^2} \)
7. \( \boxed{84\pi \, \text{cm}^2} \)
8. \( \boxed{490 \, \text{ft}^2} \)
9. \( \boxed{645 \, \text{ft}^2} \)
Parent Tip: Review the logic above to help your child master the concept of volume and surface area worksheets.
