Geometry Tutor Worksheet 17 Volume of Prisms and Pyramids

1 © MathTutorDVD.com Geometry Tutor - Worksheet 17 – Volume of Prisms and Pyramids

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

2. What is the volume 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 volume is 216 m3 whose length is 6 m, and height is 9 m?

2 © MathTutorDVD.com 4. What is the height of a rectangular prism whose volume is 672 cm3 whose length is 12 cm, and width is 7 cm?

5. What is the volume of this rectangular prism?

6. What is the volume of this cubic prism?

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

8. What is the volume of this triangular prism?

4 © MathTutorDVD.com 9. What is the volume of this rectangular prism?

10. What is the volume of this rectangular prism?

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

12. What is the volume of this triangular prism?

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

14. What is the volume of this triangular prism?

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

16. What is the volume of a square pyramid measuring 13 m along each edge of the base with a height of 12 m?

17. What is the volume of a pyramid that is 8 cm tall with a right triangle for a base with sides 18 cm, 24 cm, and 30 cm?

8 © MathTutorDVD.com 18. What is the volume of this pyramid?

19. What is the volume of this pyramid?

9 © MathTutorDVD.com 20. What is the volume of this pyramid?

21. What is the volume of this pyramid if the area of the pentagonal base of the pyramid is 249 cm2?

10 © MathTutorDVD.com 22. What is the volume of this pyramid if the area of the hexagonal base of the pyramid is 93.6 m2?

23. What is the volume of this pyramid?

11 © MathTutorDVD.com 24. What is the volume of this pyramid if the area of the pentagonal base of the pyramid is 139.5 dm2?

25. What is the volume of this pyramid?

12 © MathTutorDVD.com 26. The volume of a square pyramid is 864 m3. The height of the pyramid is 18 m. What is the length of the edges of the base?

27. The volume of a rectangular pyramid is 1080 cm3. What is the height of the pyramid if the lengths of the edges of the base are 12 cm and 18 cm?

28. The volume of a square pyramid is 2048 mm3. What is the height of the pyramid if the length of the edges of the base are 16 mm?

13 © MathTutorDVD.com Answers - Geometry Tutor - Worksheet 17 – Volume of Prisms and Pyramids

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

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

푉 = 9 m ∙ 5 m ∙ 12 m

푉 = (9 ∙ 5 ∙ 12) m3

푉 = 540 m3

Answer: 540 m3

2. What is the volume 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 volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

푉 = 4.6 cm ∙ 3.8 cm ∙ 1.4 cm

푉 = (4.6 ∙ 3.8 ∙ 1.4) cm3

푉 = 24.472 cm3

Answer: 24.472 cm3

14 © MathTutorDVD.com 3. What is the width of a rectangular prism whose volume is 216 m3 whose length is 6 m, and height is 9 m?

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

216 m3 = (6 m ∙ 푤 m ∙ 9 m)

216 m3 = (6 ∙ 푤 ∙ 9) m3

216 = 54푤

푤 = 4

Answer: 4 m

4. What is the height of a rectangular prism whose volume is 672 cm3 whose length is 12 cm, and width is 7 cm?

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

672 cm3 = (12 cm ∙ 7 cm ∙ ℎ cm)

672 cm3 = (12 ∙ 7 ∙ ℎ) cm3

672 = 84ℎ

ℎ = 8

Answer: 8 cm

15 © MathTutorDVD.com 5. What is the volume of this rectangular prism?

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

푉 = (7 m ∙ 6 m ∙ 8 m)

푉 = (7 ∙ 6 ∙ 8) m3

푉 = 336 m3

Answer: 336 m3

6. What is the volume of this cubic prism?

The formula for calculating the volume of a cubic prism is

16 © MathTutorDVD.com 푉 = 푙3 where 푉 is the volume, 푙 is the length of the sides. Therefore,

푉 = (9 cm)3

푉 = (93) cm3

푉 = 729 cm3

Answer: 729 cm3

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

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

푉 = (15 mm ∙ 9 mm ∙ 6 mm)

푉 = (15 ∙ 9 ∙ 6) mm3

푉 = 810 mm3

Answer: 810 mm3

17 © MathTutorDVD.com

8. What is the volume of this triangular prism?

Calculate the volume of a triangular prism by calculating the area of the base and multiplying that area by the length of the prism.

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

Now multiply this area by the length of the prism to find the volume.

푉 = 60 mm2 ∙ 28 mm

푉 = (60 ∙ 28) mm2 ∙ mm

푉 = 1680 mm3

Answer: 1680 mm3

18 © MathTutorDVD.com 9. What is the volume of this rectangular prism?

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

푉 = (6 km ∙ 6 km ∙ 7 km)

푉 = (6 ∙ 6 ∙ 7) km3

푉 = 252 km3

Answer: 252 km3

10. What is the volume of this rectangular prism?

19 © MathTutorDVD.com The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. Therefore,

푉 = (3.8 cm ∙ 3.6 cm ∙ 4.1 cm)

푉 = (3.8 ∙ 3.6 ∙ 4.1) cm3

푉 = 56.088 cm3

Answer: 푉 = 56.088 cm3

11. What is the volume of this rectangular prism if 퐴퐵 = 7 cm, 퐵퐶 = 11 cm, and 퐶퐺 = 14 cm?

The formula for calculating the volume of a rectangular prism is

푉 = 푙푤ℎ where 푉 is the volume, 푙 is the length, ℎ is the height, and 푤 is the width. The question gives the length, width, and height. Therefore,

푉 = (11 cm ∙ 7 cm ∙ 14 cm)

푉 = (11 ∙ 7 ∙ 14) cm3

푉 = 1078 cm3

Answer: 1078 cm3

20 © MathTutorDVD.com 12. What is the volume of this triangular prism?

Calculate the volume of a triangular prism by calculating the area of the base and multiplying that area by the length of the prism.

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

Now multiply the area of the base by the length of the prism to find the volume.

푉 = 24 m2 ∙ 16 m

푉 = (24 ∙ 16) m2 ∙ m

푉 = 384 m3

Answer: 384 m3

21 © MathTutorDVD.com 13. What is the volume of this heptagonal prism if the area of the base is 140 mm2?

Calculate the volume of a heptagonal prism by multiplying the area of the base by the height of the prism.

The area of the bases is given as 140 mm2.

Now multiply the area of the base by the length of the prism to find the volume.

푉 = 140 mm2 ∙ 8 mm

푉 = (140 ∙ 8) mm2 ∙ mm

퐴 = 1120 mm3

Answer: 1120 mm3

22 © MathTutorDVD.com 14. What is the volume of this triangular prism?

Calculate the volume of a triangular prism by calculating the area of the base and multiplying that area by the length of the prism.

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

퐵 = 9√3 dm2

Now multiply the area of the base by the length of the prism to find the volume.

푉 = 9√3 dm2 ∙ 13 dm

푉 = (9√3 ∙ 13)dm2 ∙ dm

푉 = 117√3 dm3

Answer: 117√3 dm3

Calculate the volume of a regular pentagonal prism by multiplying the area of the base by the height or length of the prism.

The area of the base is given as 30 hm2. Multiply this area by the length of the prism.

푉 = 30 hm2 ∙ 15 hm

푉 = (30 ∙ 15) hm2 ∙ hm

푉 = 450 hm3

Answer: 450 hm3

16. What is the volume of a square pyramid measuring 13 m along each edge of the base with a height of 12 m?

1 The formula for calculating the volume of a pyramid is 푉 = 퐵ℎ where 푉 is the 3 volume, 퐵 is the area of the base, and ℎ is the height of the pyramid. As such, the value for 퐵 is calculated using the area formula for the base of the pyramid.

In this pyramid, the base is a square, so 퐵 = 푠2 = (13 m)2 = 169 m2.

Therefore, the volume is

17. What is the volume of a triangular pyramid that is 8 cm tall with a right triangle for a base with sides 18 cm, 24 cm, and 30 cm?

In this pyramid, the base is a right triangle with legs 18 cm and 24 cm, so 1 1 퐵 = 푏ℎ = ∙ 18 cm ∙ 24 cm = 216 cm2. 2 2 Therefore, the volume is 1 1 푉 = (216 cm2 ∙ 8 cm) = (216 ∙ 8) cm2 ∙ cm = 576 cm3 3 3 Answer: 576 cm3

18. What is the volume of this pyramid?

25 © MathTutorDVD.com 1 In this pyramid, the base is a right triangle with legs 3 m and 4 m, so 퐵 = 푏ℎ = 2 1 ∙ 3 m ∙ 4 m = 6 m2. 2 Therefore, the volume is 1 1 푉 = (6 m2 ∙ 4 m) = (6 ∙ 4) m2 ∙ m = 8 m3 3 3 Answer: 8 m3

19. What is the volume of this pyramid?

In this pyramid, the base is a rectangle with sides 11 cm and 10 cm, so 퐵 = 푏ℎ = 11 cm ∙ 10 cm = 110 cm2.

Therefore, the volume is 1 1 푉 = (110 cm2 ∙ 12 cm) = (110 ∙ 12) cm2 ∙ cm = 440 cm3 3 3 Answer: 440 cm3

In this pyramid, the base is a rectangle with sides 3 m and 12 m, so 퐵 = 푏ℎ = 3 m ∙ 12 m = 36 m2.

Therefore, the volume is 1 1 푉 = (36 m2 ∙ 13 m) = (36 ∙ 13) m2 ∙ m = 156 m3 3 3 Answer: 156 m3

21. What is the volume of this pyramid if the area of the pentagonal base of the pyramid is 249 cm2?

27 © MathTutorDVD.com 1 The formula for calculating the volume of a pyramid is 푉 = 퐵ℎ where 푉 is the 3 volume, 퐵 is the area of the base, and ℎ is the height of the pyramid. As such, the value for 퐵 is calculated using the area formula for the base of the pyramid.

In this pyramid, the area of the pentagonal base is provided as 퐵 = 249 cm2.

Therefore, the volume is 1 1 푉 = (249 cm2 ∙ 11 cm) = (249 ∙ 11) cm2 ∙ cm = 913 cm3 3 3 Answer: 913 cm3

22. What is the volume of this pyramid if the area of the hexagonal base of the pyramid is 93.6 m2?

In this pyramid, the area of the hexagonal base is provided as 퐵 = 93.6 m2.

Therefore, the volume is 1 1 푉 = (93.6 m2 ∙ 9 m) = (93.6 ∙ 9) m2 ∙ m = 280.8 m3 3 3 Answer: 280.8 m3

In this pyramid, the base is a right triangle with legs 6 mm and 8 mm, so 1 1 퐵 = 푏ℎ = ∙ 6 mm ∙ 8 mm = 24 mm2. 2 2 Therefore, the volume is 1 1 푉 = (24 mm2 ∙ 12 mm) = (24 ∙ 12) mm2 ∙ mm = 96 mm3 3 3 Answer: 96 mm3

24. What is the volume of this pyramid if the area of the pentagonal base of the pyramid is 139.5 dm2?

29 © MathTutorDVD.com 1 The formula for calculating the volume of a pyramid is 푉 = 퐵ℎ where 푉 is the 3 volume, 퐵 is the area of the base, and ℎ is the height of the pyramid. As such, the value for 퐵 is calculated using the area formula for the base of the pyramid.

In this pyramid, the area of the pentagonal base is provided as 퐵 = 139.5 dm2.

Therefore, the volume is 1 1 푉 = (139.5 dm2 ∙ 7 dm) = (139.5 ∙ 7) dm2 ∙ dm = 325.5 dm3 3 3 Answer: 325.5 dm3

25. What is the volume of this pyramid?

In this pyramid, the base is a rectangle with sides 8 in and 10 in, so 퐵 = 푏ℎ = 8 in ∙ 10 in = 80 in2.

Therefore, the volume is 1 1 푉 = (80 in2 ∙ 15 in) = (80 ∙ 15) in2 ∙ in = 400 in3 3 3 Answer: 400 in3

30 © MathTutorDVD.com 26. The volume of a square pyramid is 864 m3. The height of the pyramid is 18 m. What is the length of the edges of the base?

In this pyramid, the base is a square, so 퐵 = 푠2. Use the volume formula to find the length of the edges of the base.

Therefore, 1 864 m3 = (푠2 m2 ∙ 18 m) 3 1 864 m3 = (푠2 ∙ 18 ) m3 3 864 = 6푠2

144 = 푠2

푠 = 12

Answer: 12 m

27. The volume of a rectangular pyramid is 1080 cm3. What is the height of the pyramid if the lengths of the edges of the base are 12 cm and 18 cm?

In this pyramid, the base is a rectangle with sides 12 cm and 18 cm, so 퐵 = 푏ℎ = 12 cm ∙ 18 cm = 216 cm2.

Use the volume formula to find ℎ. 1 1080 cm3 = (216 cm2 ∙ ℎ cm) 3 31 © MathTutorDVD.com 1 1080 cm3 = (216 ∙ ℎ) cm3 3 1080 = 72ℎ

ℎ = 15

Answer: 15 cm

28. The volume of a square pyramid is 2048 mm3. What is the height of the pyramid if the length of the edges of the base are 16 mm?

In this pyramid, the base is a square, so 퐵 = 푠2 = 162 = 256 mm2. Use the volume formula to find the length of the height.

Therefore, 1 2048 mm3 = (256 mm2 ∙ ℎ mm) 3 1 2048 mm3 = (256 ∙ ℎ ) mm3 3 6144 = 256ℎ

24 = ℎ

Answer: 24 mm