Volume of Pyramids and Cones Worksheet

**Page 2**

11.

If the side of each cube in the shown pyramid is 4 cm, then find the total volume occupied by the cubes.

a. | 7676 cm ^{3} | ||

b. | 7670 cm ^{3} | ||

c. | 7680 cm ^{3} | ||

d. | 7665 cm ^{3} |

Number of cubes in the first layer = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36

Number of cubes in the second layer = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28

Number of cubes in the third layer = 6 + 5 + 4 + 3 + 2 + 1 = 21

Number of cubes in the fourth layer = 5 + 4 + 3 + 2 + 1 = 15

Number of cubes in the fifth layer = 4 + 3 + 2 + 1 = 10

Number of cubes in the sixth layer = 3 + 2 + 1 = 6

Number of cubes in the seventh layer = 2 + 1 = 3

Number of cubes in the eighth layer = 1

So, total number of cubes in the pyramid = 36 + 28 + 21 + 15 + 10 + 6 + 3 + 1 = 120

Volume of each cube = 64 cm

[Side of each cube = 4 cm.]

The volume occupied by the cubes of the pyramid = Total number of cubes × volume of each cube = 120 × 64 = 7680 cm

Correct answer : (3)

12.

Find the volume of a cone and round the answer to the nearest whole unit.

a. | 2366.3 in ^{3} | ||

b. | 2366 in ^{3} | ||

c. | 2365 in ^{3} | ||

d. | 2377 in ^{3} |

[Formulae.]

=

[ r =10.5; h = 20.5 and substituting the values.]

= 2366.3 in.

[ Simplify and round the answer to the nearest whole unit.]

The volume of the cone to the nearest whole unit is 2366 in.

Correct answer : (2)

13.

Find the volume of a cone, if the diameter of the cone is 12.4 cm. and the height of the cone is 30 cm.

a. | 1206 $\mathrm{cm}$ ^{3} | ||

b. | 1207.4 $\mathrm{cm}$ ^{3} | ||

c. | 1207 $\mathrm{cm}$ ^{3} | ||

d. | 1208 $\mathrm{cm}$ ^{3} |

[Formulae.]

r = 6.2 cm.; h = 30 cm.

[ Radius is half of diameter.]

V =

[Substituting the values.]

= 1207.4 cm.

[Simplify.]

Therefore, the volume of the cone is 1207.4 cm.

Correct answer : (2)

14.

Find the volume of the pyramid.

a. | 52.26 cm ^{3} | ||

b. | 52 cm ^{3} | ||

c. | 53 cm ^{3} | ||

d. | 51 cm ^{3} |

[Formulae.]

V =

[Here, B = s × s (i.e., area of a square) and substituting the values.]

= 52.26 cm.

[Simplify.]

So, the volume of the pyramid is 52.26 cm.

Correct answer : (1)

15.

Find the volume of the figure given.

a. | 314 π in ^{3} | ||

b. | 110 π in ^{3} | ||

c. | 3.14 π in ^{3} | ||

d. | 100 π in ^{3} |

[Formula.]

Here, r = 5 in.; h = 6 + 6 = 12 in.

V =

[Substituting the values.]

= 100 π in.

[Simplify.]

The volume of a given figure is 100 π in.

Correct answer : (4)

16.

Base radius of a cone is 5 in. and its height is 15 in. If the height and the radius of the cone are doubled, then find the volume of the new cone.[Use $\pi $ = 3.]

a. | 1500 in. ^{3} | ||

b. | 9000 in. ^{3} | ||

c. | 900 in. ^{3} | ||

d. | 3000 in. ^{3} |

[Since base radius of new cone is double the radius of initial cone.]

Height of the new cone = 2 × height of the initial cone = 2 × 15 = 30 in.

[Since height of new cone is double the height of initial cone.]

Volume of cone =

[Formula.]

= 3000

Volume of new cone =

[Substitute the values in the formula and simplify.]

Volume of the new cone is 3000 in.

Correct answer : (4)

17.

The circumference of the base of a cone of height 15 in. is 44 in. Find the volume of the cone. (Use π = $\frac{22}{7}$)

a. | 770 in. ^{3} | ||

b. | 766 in. ^{3} | ||

c. | 784 in. ^{3} | ||

d. | 774 in. ^{3} |

[Formula.]

2π

[Since circumference of a cone is 44 in.]

2 ×

[Simplify.]

Volume of the cone =

[Formula.]

=

[Substitute the values.]

= 770

[Simplify.]

Volume of the cone = 770 in.

Correct answer : (1)

18.

A bucket is in the shape of a frustum of a cone with a height($x$) of 12 cm, diameter of the top portion($a$) 48 cm and diameter of bottom portion($b$) 16 cm. Find the capacity of the bucket. [Take $\pi $ = 3.]

a. | 29952 cm ^{3} | ||

b. | 2688 cm ^{3} | ||

c. | 9984 cm ^{3} | ||

d. | 2304 cm ^{3} |

The bucket can be considered as a cone with the upper part removed.

[Analysis.]

ΔFBA represents the full cone, ΔEBD represents the part of the cone removed.

[From the figure.]

ΔOAB ~ ΔCDB.

[From the figure.]

[From step 3.]

[Substitute.]

[Simplify.]

The volume of the bucket = Volume of the cone with base diameter 48 cm - Volume of the cone with base diameter 16 cm

[Formula.]

Height of cone with diameter 48 cm = 6 + 12

Height of cone with diameter 16 cm = 6 cm

Volume of the cone with diameter 48 cm =

[Volume of the cone =

Volume of the cone with diameter 16 cm =

[Formula.]

Volume of the bucket = 10368 - 384

[Simplify.]

Capacity of the bucket = 9984 cm

Correct answer : (3)

19.

The circumference of the base of a cone of height 18 in. is 44 in. Find the volume of the cone.

a. | 939.18 in. ^{3} | ||

b. | 924.38 in. ^{3} | ||

c. | 919.58 in. ^{3} | ||

d. | 929.18 in. ^{3} |

[Formula.]

2π

[Since circumference of a cone is 44 in.]

2 ×

[Simplify.]

Volume of the cone =

[Formula.]

= 924.38

[Simplify.]

Volume of the cone = 924.38 in.

Correct answer : (2)

20.

Which of the following statements is true?

a. | The volume of a pyramid is $\frac{1}{3}$ times the product of base area and height. | ||

b. | The volume of a prism is $\frac{1}{3}$ times the sum of base area and height. | ||

c. | The volume of a pyramid is $\frac{1}{2}$ times the product of base area and height. | ||

d. | The volume of a prism is $\frac{1}{2}$ times the product of base area and its height. |

So, the statement, 'The volume of a pyramid is

Correct answer : (1)