# 45 Answer Key Surface Area Of Prisms And Pyramids Worksheet Answers

## Introduction

Welcome to our blog post on the answer key for the surface area of prisms and pyramids worksheet. In this article, we will provide you with the answers to the worksheet questions, allowing you to check your work and ensure that you have a thorough understanding of these important geometric concepts. We will break down the answers by category, covering both prisms and pyramids separately. So, let's dive right in and explore the answer key!

## Prisms

### Rectangular Prism

Question 1: Find the surface area of a rectangular prism with dimensions 5 cm, 6 cm, and 8 cm.

Answer: The surface area of a rectangular prism is given by the formula: 2lw + 2lh + 2wh. Plugging in the values, we get: 2(5)(6) + 2(5)(8) + 2(6)(8) = 60 + 80 + 96 = 236 cm².

### Square Prism

Question 2: Determine the surface area of a square prism with edge length 4 cm.

Answer: For a square prism, the surface area formula is: 6s², where s is the length of the edge. Substituting the value, we have: 6(4²) = 6(16) = 96 cm².

### Triangular Prism

Question 3: Calculate the surface area of a triangular prism with base sides measuring 5 cm, 6 cm, and 7 cm, and a height of 8 cm.

Answer: The formula for the surface area of a triangular prism is: (2 × base area) + (base perimeter × height). First, we find the base area using Heron's formula: √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter and a, b, and c are the side lengths. Plugging in the values, we get: s = (5 + 6 + 7)/2 = 9. a = 5, b = 6, c = 7. Therefore, the base area is √(9(9-5)(9-6)(9-7)) = √(9(4)(3)(2)) = √(9(24)) = √216 = 14.7 cm². Next, we calculate the base perimeter: 5 + 6 + 7 = 18 cm. Finally, we substitute these values into the formula: (2 × 14.7) + (18 × 8) = 29.4 + 144 = 173.4 cm².

### Pentagonal Prism

Question 4: Find the surface area of a pentagonal prism with a base side length of 3 cm and a height of 10 cm.

Answer: The surface area of a pentagonal prism can be determined using the formula: (5 × base area) + (base perimeter × height). To find the base area, we need to compute the apothem length (the distance from the center of the pentagon to the midpoint of a side) and the perimeter. The apothem is given by the formula: (side length)/(2 × tan(180°/n)), where n is the number of sides. Therefore, the apothem is (3)/(2 × tan(180°/5)) = 1.853 cm. The perimeter of the base is 5 × (side length) = 5 × 3 = 15 cm. Plugging these values into the surface area formula, we get: (5 × (3 × 1.853)) + (15 × 10) = 27.795 + 150 = 177.795 cm².

## Pyramids

### Rectangular Pyramid

Question 5: Determine the surface area of a rectangular pyramid with base dimensions of 6 cm and 8 cm, and a slant height of 10 cm.

Answer: The surface area of a rectangular pyramid is given by the formula: (base area) + (½ × perimeter × slant height). First, we find the base area: 6 × 8 = 48 cm². Next, we calculate the perimeter of the base: 2(6 + 8) = 28 cm. Finally, we substitute these values into the formula: 48 + (½ × 28 × 10) = 48 + (14 × 10) = 48 + 140 = 188 cm².

### Square Pyramid

Question 6: Find the surface area of a square pyramid with a base side length of 5 cm and a slant height of 7 cm.

Answer: The surface area of a square pyramid can be determined using the formula: (base area) + (½ × perimeter × slant height). First, we find the base area: 5² = 25 cm². Next, we calculate the perimeter of the base: 4 × 5 = 20 cm. Finally, we substitute these values into the formula: 25 + (½ × 20 × 7) = 25 + (10 × 7) = 25 + 70 = 95 cm².

### Triangular Pyramid

Question 7: Calculate the surface area of a triangular pyramid with base sides measuring 4 cm, 5 cm, and 6 cm, and a slant height of 8 cm.

Answer: The surface area of a triangular pyramid is given by the formula: (base area) + (½ × perimeter × slant height). First, we find the base area using Heron's formula: √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter and a, b, and c are the side lengths. Plugging in the values, we get: s = (4 + 5 + 6)/2 = 7.5. a = 4, b = 5, c = 6. Therefore, the base area is √(7.5(7.5-4)(7.5-5)(7.5-6)) = √(7.5(3.5)(2.5)(1.5)) = √(7.5(13.125)) = √98.4375 = 9.921 cm². Next, we calculate the perimeter of the base: 4 + 5 + 6 = 15 cm. Finally, we substitute these values into the formula: 9.921 + (½ × 15 × 8) = 9.921 + (7.5 × 8) = 9.921 + 60 = 69.921 cm².

### Pentagonal Pyramid

Question 8: Find the surface area of a pentagonal pyramid with a base side length of 3 cm and a slant height of 9 cm.

Answer: The surface area of a pentagonal pyramid can be determined using the formula: (base area) + (½ × perimeter × slant height). To find the base area, we need to compute the apothem length (the distance from the center of the pentagon to the midpoint of a side) and the perimeter. The apothem is given by the formula: (side length)/(2 × tan(180°/n)), where n is the number of sides. Therefore, the apothem is (3)/(2 × tan(180°/5)) = 1.853 cm. The perimeter of the base is 5 × (side length) = 5 × 3 = 15 cm. Plugging these values into the surface area formula, we get: (½ × (3 × 15)) + (½ × 15 × 9) = 22.5 + 67.5 = 90 cm².

## Conclusion

That brings us to the end of our answer key for the surface area of prisms and pyramids worksheet. We hope that this article has provided you with a clear understanding of how to calculate the surface area of various geometric shapes. Remember to always double-check your work and use the appropriate formulas for each shape. If you have any further questions or need additional clarification, don't hesitate to reach out to your teacher or instructor. Happy calculating!