PRACTICE PROBLEMS ON VOLUME OF 3D SHAPES

Problem 1 :

Find the volume of the cube

Solution :

Side length of the cube = 6 cm

Volume of cube = a³

= 6³

= 216 cm³

Problem 2 :

A cuboid has a length, width and height of 4 cm, 5 cm and 9 cm, respectively. Find the volume of the cuboid.

Solution :

Volume of cuboid = base area x height

length = 4 cm, width = 5 cm and height = 9 cm

= (4 x 5) x 9

= 20 x 9

= 180 cm³

Problem 3 :

Find the width of a cuboid, given that it has a length of 7 cm, height of 10 cm and volume of 490 cm³

Solution :

Volume = 490 cm³

length x width x height = 490 cm³

length = 7 cm and height = 10 cm

7 x w x 10 = 490

w = 490/(7 x 10)

w = 7 cm

So, the width is 7 cm.

Problem 4 :

Find its length, given that the volume of a cube is 343 cm³ 

Solution :

Side length = a

volume of cube = a³

a³ = 343

a = ∛343

a = ∛7 x 7 x 7

a = 7 cm

So, side length of the cube is 7 cm.

Problem 5 :

Find the volume of the triangular prism 

Solution :

Volume of the triangular prism = Area of triangle x height

Area of triangle = (1/2) x base x height

= (1/2) x 4 x 5

= 10 cm²

Volume of the triangular prism = 10 x 12

= 120 cm²

Problem 6 :

Find the volume of the trapezoidal prism

Solution :

Volume of trapezoidal prism = Area of trapezium x height

Area of trapezium = (1/2) x (sum of parallel sides) x height

= (1/2) (4 + 7) x 5

= (1/2) x 11 x 5

= 27.5 cm²

Volume of trapezoidal prism = 27.5 x 10

= 275 cm²

Problem 7 :

Find the volume of the square-based pyramid, rounding your answer to 3 significant figures

Solution :

Volume of pyramid = 1/3 x area of base x height

(1/3) x 5 x 5 x 4

= (1/3) x 100

= 33.3 cm³

Problem 8 :

Find the volume of the tetrahedron, rounding your answer to 3 significant figures

Solution :

Area of base = (√3/4) x a²

= (√3/4) x (10)²

= 25√3

Volume of tetrahedron = (1/3) x 25√3 x 8

= 115.46 cm³

Problem 9 :

Find the volume of the cone, rounding your answer to 3 significant figures

Solution :

Base area = πr²

= π(3)²

= 9π

Volume of cone = (1/3) x Base area x height

= (1/3) x 9π x 8

= 24π

= 24(3.14)

= 75.36

= 75.4 cm³

Problem 10 :

Find the volume of the sphere, rounding your answer to 3 significant figures

Solution :

Volume = (1/3) x base area x height

= (1/3)(4πr²) x r

= (1/3)(4π(6)²) x (6)

= (864/3)π

= 288π

= 904.32

= 904 cm³

Problem 11 :

A hemisphere has a radius of 2 cm. Find its volume, rounding your answer to 3 significant figures. 

Solution :

Volume = (1/3) x base area x height

= (1/3)(2πr²) x r

= (1/3)(2π(2)²) x (2)

= (16/3)π

= 5.3(3.14)

= 16.74 cm³

Problem 12 :

The solid shown below is a cuboid with a square-based pyramid on top. The pyramid has a vertical height of 4 cm. Find the volume of the solid, giving your answer to one decimal place where necessary.

Solution :

Volume of the given shape = volume of top + volume of bottom

= 9 x 9 x 7 + (1/3)(9 x 9) x 4

= 567 + 108

= 675 cm³

Problem 13 :

A tank on the road roller is filled with water to make the roller heavy. The tank is a cylinder that has a height of 6 feet and a radius of 2 feet. One cubic foot of water weighs 62.5 pounds. Find the weight of the water in the tank

Solution :

height of the tank = 6 feet, radius = 2 feet

Volume v = πr²h

= 3.14 × (2)² × 6

= 3.14 × 4 × 6

= 75.36 ft

So, the weight of the water in the tank is 75.36 ft.

One cubic foot of water weighs = 62.5 pounds

= 62.5 × 75.36

= 4710 pounds.

Problem 14 :

Water flows at 2 feet per second through a pipe with a diameter of 8 inches. A cylindrical tank with a diameter of 15 feet and a height of 6 feet collects the water.

a) what is the volume, in cubic inches, of water flowing out of the pipe every second.

b) What is the height, in inches, of the water in the tank after 5 minutes?

c) How many minutes will it take to fill 75% of the tank?

Solution :

Diameter d = 8 feet ==> radius = 4 feet

height (h) = 2 feet ==> 24 inches

(a)  Volume of water flows out every second = πr²h

=  π x 4² x 24

= 384π

Using π = 3.14, we get

= 1206 cubic inches

(b)  quantity of water in the cylindrical tank after 5 minutes : 

5 minutes = 300 seconds

Quantity of water = 384π x 300

= 115200π

Volume of water in cylindrical tank = 115200π

πr²h = 115200π

r = 15/2 ==> 7.5 feet ==> 90 inches

π(90)² h = 115200π

h = 115200/8100

h = 14.2 inches

c) How many minutes will it take to fill 75% of the tank?

Capacity of the tank = πr²h

r = 90 inches, h = 6x12 ==> 72 inches

= π(90)²(72)

= 583200π

75% of the capacity = 0.75 (583200π)

= 437400π

Volume of water in 1 second = 384π

= 437400π/384π

= 1140 seconds

Converting seconds to minutes,

= 1140/60

= 19 minutes