Geometry Tutor Worksheet 16 Area of Prisms

1 © MathTutorDVD.com Geometry Tutor - Worksheet 16 – Area of Prisms

1. What is the total surface area of a rectangular prism whose length is 9 m, width is 5 m, and height is 12 m?

2. What is the total surface area of a rectangular prism whose length is 4.6 cm, width is 3.8 cm, and height is 1.4 cm?

3. What is the width of a rectangular prism whose total surface area is 184 m2 whose length is 5 m, and height is 4 m?

2 © MathTutorDVD.com 4. What is the height of a rectangular prism whose total surface area is 318 cm2 whose length is 6 cm, and width is 9 cm?

5. What is the total surface area of this rectangular prism?

6. What is the total surface area of this cubic prism?

3 © MathTutorDVD.com 7. What is the total surface area of this rectangular prism if 푈푉 = 9 mm, 푆푌 = 15 mm, and 푇푊 = 6 mm?

8. What is the total surface area of this triangular prism?

4 © MathTutorDVD.com 9. What is the total surface area of this rectangular prism?

10. What is the total surface area of this rectangular prism?

5 © MathTutorDVD.com 11. What is the total surface area of this rectangular prism if 퐴퐵 = 7 cm, 퐵퐶 = 11 cm, and 퐶퐺 = 14 cm?

12. What is the total surface area of this triangular prism?

6 © MathTutorDVD.com 13. What is the total surface area of this heptagonal prism if the area of the base is 140 mm2?

14. What is the total surface area of this triangular prism?

7 © MathTutorDVD.com 15. What is the total surface area of this regular pentagonal prism if the area of the base is 30 hm2?

8 © MathTutorDVD.com Answers - Geometry Tutor - Worksheet 16 – Area of Prisms

1. What is the total surface area of a rectangular prism whose length is 9 m, width is 5 m, and height is 12 m?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

퐴 = 2(9 m ∙ 12 m + 9 m ∙ 5 m + 12 m ∙ 5 m)

퐴 = 2(9 ∙ 12 + 9 ∙ 5 + 12 ∙ 5) m2

퐴 = 2(213 ) m2

퐴 = 426 m2

Answer: 426 m2

2. What is the total surface area of a rectangular prism whose length is 4.6 cm, width is 3.8 cm, and height is 1.4 cm?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

퐴 = 2(4.6 cm ∙ 1.4 cm + 4.6 cm ∙ 3.8 cm + 1.4 cm ∙ 3.8 cm)

퐴 = 2(4.6 ∙ 1.4 + 4.6 ∙ 3.8 + 1.4 ∙ 3.8) cm2

퐴 = 2(29.24 ) cm2

퐴 = 58.48 cm2

Answer: 58.48 cm2

9 © MathTutorDVD.com 3. What is the width of a rectangular prism whose total surface area is 184 m2 whose length is 5 m, and height is 4 m?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

184 m2 = 2(5 m ∙ 4 m + 5 m ∙ 푥 m + 4 m ∙ 푥 m)

184 m2 = 2(5 ∙ 4 + 5 ∙ 푥 + 4 ∙ 푥) m2

184 = 2(20 + 9푥 )

92 = 20 + 9푥

72 = 9푥

푥 = 8

Answer: 8 m

4. What is the height of a rectangular prism whose total surface area is 318 cm2 whose length is 6 cm, and width is 9 cm?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

318 cm2 = 2(6 cm ∙ 푥 cm + 6 cm ∙ 9 cm + 푥 cm ∙ 9 cm)

318 cm2 = 2(6 ∙ 푥 + 6 ∙ 9 + 푥 ∙ 9) cm2

318 = 2(15푥 + 54)

159 = 15푥 + 54

105 = 15푥

푥 = 7 10 © MathTutorDVD.com Answer: 7 cm

5. What is the total surface area of this rectangular prism?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

퐴 = 2(7 m ∙ 8 m + 7 m ∙ 6 m + 8 m ∙ 6 m)

퐴 = 2(7 ∙ 8 + 7 ∙ 6 + 8 ∙ 6) m2

퐴 = 2(146) m2

퐴 = 292 m2

Answer: 292 m2

11 © MathTutorDVD.com 6. What is the total surface area of this cubic prism?

The formula for calculating the total surface area of a cubic prism is

퐴 = 6푙2 where 퐴 is the area, 푙 is the length of the sides. Therefore,

퐴 = 6 ∙ (9 cm)2

퐴 = 6(81) cm2

퐴 = 486 cm2

Answer: 486 cm2

7. What is the total surface area of this rectangular prism if 푈푉 = 9 mm, 푆푌 = 15 mm, and 푇푊 = 6 mm?

The formula for calculating the total surface area of a rectangular prism is

12 © MathTutorDVD.com 퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

퐴 = 2(15 mm ∙ 6 mm + 15 mm ∙ 9 mm + 6 mm ∙ 9 mm)

퐴 = 2(15 ∙ 6 + 15 ∙ 9 + 6 ∙ 9) mm2

퐴 = 2(279) mm2

퐴 = 558 mm2

Answer: 558 mm2

8. What is the total surface area of this triangular prism?

Calculate the total surface area of a triangular prism by calculating the area of the bases and the lateral surface areas and add the areas.

The formula for areas of the bases is twice the area of a triangle. Therefore, the 1 formula is 퐵 = 2 ( 푏ℎ) where 퐵 is the area of the bases, 푏 is the length of the 2 base of the triangles, and ℎ is the height of the triangles. Therefore, 1 퐵 = 2 ( ∙ 24 mm ∙ 5 mm) 2 1 퐵 = 2 ( ∙ 24 ∙ 5) mm2 2 13 © MathTutorDVD.com 퐵 = 2(60) mm2

퐵 = 120 mm2

Now calculate the areas of the lateral faces which are all rectangles and add all of the areas together.

퐴 = 120 mm2 + 28 mm ∙ 24 mm + 28 mm ∙ 13 mm + 28 mm ∙ 13 mm

퐴 = 120 mm2 + 28 ∙ 24 mm2 + 28 ∙ 13 mm2 + 28 ∙ 13 mm2

퐴 = 120 mm2 + 672 mm2 + 364 mm2 + 364 mm2

퐴 = 1520 mm2

Answer: 1520 mm2

9. What is the total surface area of this rectangular prism?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

퐴 = 2(6 km ∙ 7 km + 6 km ∙ 6 km + 7 km ∙ 6 km)

퐴 = 2(6 ∙ 7 + 6 ∙ 6 + 7 ∙ 6) km2

퐴 = 2(120) km2 14 © MathTutorDVD.com 퐴 = 240 km2

Answer: 240 km2

10. What is the total surface area of this rectangular prism?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

퐴 = 2(3.8 cm ∙ 4.1 cm + 3.8 cm ∙ 3.6 cm + 4.1 cm ∙ 3.6 cm)

퐴 = 2(3.8 ∙ 4.1 + 3.8 ∙ 3.6 + 4.1 ∙ 3.6) cm2

퐴 = 2(44.02 ) cm2

퐴 = 88.04 cm2

Answer: 88.04 cm2

15 © MathTutorDVD.com 11. What is the total surface area of this rectangular prism if 퐴퐵 = 7 cm, 퐵퐶 = 11 cm, and 퐶퐺 = 14 cm?

The formula for calculating the total surface area of a rectangular prism is

퐴 = 2(푙ℎ + 푙푤 + ℎ푤) where 퐴 is the area, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

퐴 = 2(11 cm ∙ 14 cm + 11 cm ∙ 7 cm + 14 cm ∙ 7 cm)

퐴 = 2(11 ∙ 14 + 11 ∙ 7 + 14 ∙ 7) cm2

퐴 = 2(329) cm2

퐴 = 658 cm2

Answer: 658 cm2

Calculate the total surface area of a triangular prism by calculating the area of the bases and the lateral surface areas and add the areas.

The formula for areas of the bases is twice the area of a triangle. Therefore, the 1 formula is 퐵 = 2 ( 푏ℎ) where 퐵 is the area of the bases, 푏 is the length of the 2 base of the triangles, and ℎ is the height of the triangles. The triangles are right triangles, so the legs are the base and the height. Therefore, 1 퐵 = 2 ( ∙ 6 m ∙ 6 m) 2 1 퐵 = 2 ( ∙ 6 ∙ 8) m2 2 퐵 = 2(24) m2

퐵 = 48 m2

Now calculate the areas of the lateral faces which are all rectangles and add all of the areas together. Notice again that the triangular bases are right triangles, so using the Pythagorean Theorem formula, we calculate that the length of the hypotenuse is 10 cm.

퐴 = 48 m2 + 8 m ∙ 16 m + 6 m ∙ 16 m + 10 m ∙ 16 m

퐴 = 48 m2 + 8 ∙ 16 m2 + 6 ∙ 16 m2 + 10 ∙ 16 m2

퐴 = 48 m2 + 128 m2 + 96 m2 + 160 m2

Answer: 432 m2

13. What is the total surface area of this heptagonal prism if the area of the base is 140 mm2?

Calculate the total surface area of a heptagonal prism by calculating the area of the bases and the lateral surface areas and add the areas.

The area of the bases is given as 퐵 = 140 mm2 where 퐵 is the area of one base, so the area of both bases is 2퐵 = 2 ∙ 140 mm2 = 280 mm2.

Now calculate the areas of the lateral faces which are all rectangles and add all of the areas together. Since some side lengths are equal to each other, we can combine them.

퐴 = 280 mm2 + 2(4 mm ∙ 8 mm) + 7 mm ∙ 8 mm + 2(6 mm ∙ 8 mm) + 2(8 mm ∙ 8 mm)

퐴 = 280 mm2 + 2(4 ∙ 8 ) mm2 + 7 ∙ 8 mm2 + 2(6 ∙ 8 ) mm2 + 2(8 ∙ 8 ) mm2

퐴 = 280 mm2 + 64 mm2 + 56 mm2 + 96 mm2 + 128 mm2

퐴 = 624 mm2

Answer: 624 mm2

Calculate the total surface area of a triangular prism by calculating the area of the bases and the lateral surface areas and add the areas.

The formula for areas of the bases is twice the area of a triangle. Therefore, the 1 formula is 퐵 = 2 ( 푏ℎ) where 퐵 is the area of the bases, 푏 is the length of the 2 base of the triangles, and ℎ is the height of the triangles. The triangles at the 1 bases are equilateral triangles, whose height is calculated as (6)(√3) = 3√3. 2 Therefore, 1 퐵 = 2 ( ∙ 6 dm ∙ 3√3 dm) 2 1 퐵 = 2 ( ∙ 6 ∙ 3√3) dm2 2

퐵 = 2(9√3) dm2

퐵 = 18√3 dm2

Now calculate the areas of the lateral faces which are all rectangles and add all of the areas together. Since the triangles are equilateral, we can combine the lateral faces because they are equal to each other.

퐴 = 18√3 dm2 + 3(6 dm ∙ 13 dm)

Answer: 18√3 dm2 + 234 dm2 or (18√3 + 234) dm2

15. What is the total surface area of this regular pentagonal prism if the area of the base is 30 hm2?

Calculate the total surface area of a regular pentagonal prism by calculating the area of the bases and the lateral surface areas and add the areas.

The area of the bases is given as 퐵 = 30 hm2 where 퐵 is the area of one base, so the area of both bases is 2퐵 = 2 ∙ 30 hm2 = 60 hm2.

Now calculate the areas of the lateral faces which are all rectangles and add all of the areas together. Since all side lengths are equal to each other, we can combine them.

퐴 = 60 hm2 + 5(4 hm ∙ 15 hm)

퐴 = 60 hm2 + 5(4 ∙ 15 ) hm2

퐴 = 60 hm2 + 300 hm2

퐴 = 360 hm2

Answer: 360 hm2