Surface Area of Pyramids and Cones Worksheet

**Page 1**

1.

What is the base area of the square pyramid whose lateral edge is 5 cm. and the height is 4 cm.?

a. | 9 cm ^{2} | ||

b. | 18 cm ^{2} | ||

c. | 36 cm ^{2} | ||

d. | 10 cm ^{2} |

[Height AB is perpendicular to BC and the lateral edge AC is the hypotenuse.]

BC

[Apply Pythagorean theorem.]

BC

[Substitute height, AB = 4 and lateral edge, AC = 5.]

BC

[Simplify.]

BC

[Subtract 16 from both sides.]

BC =

[Take square root on both sides.]

Diagonal of the square base = 2 × BC

[BC is half the diagonal of the square base.]

= 2 × 3 = 6 cm.

[Substitute and multiply.]

Area of a square =

[Formula for the area of a square in terms of the measure of diagonals.]

=

[The diagonals of a square have equal measures. Substitute the values.]

The base area of the square pyramid is 18 cm.

Correct answer : (2)

2.

The circumference of the base of a conical tent is 50.24 m and its slant height is 12 m. Find the area of the canvas used in making the tent.

a. | 298.44 m ^{2} | ||

b. | 304.44 m ^{2} | ||

c. | 301.44 m ^{2} | ||

d. | 307.44 m ^{2} |

Circumference of the conical tent = 2

2 × π ×

Lateral area of the tent =

[Formula.]

= π ×

[Substitute the values.]

= 301.44 m

[Multiply.]

So, area of the canvas required is 301.44 m

Correct answer : (3)

3.

Find the slant height of the cone if its lateral area is 516 ft^{2} and its radius is 12 ft. (Round the answer to the nearest hundredth) Use π = 3.14 .

a. | 135.02 ft | ||

b. | 15.69 ft | ||

c. | 13.69 ft | ||

d. | 11.69 ft |

Radius of the cone = 12 ft

π × 12 ×

[Substitute the value of

[Divide each side by 12.]

π

[Simplify.]

3.14 ×

[Substitute π = 3.14.]

[Divide each side 3.14.]

[Simplify.]

The slant height of the cone is 13.69 ft.

Correct answer : (3)

4.

The height(h) of a cone is 40 cm. and its diameter(d) is 60 cm. What is its slant height?

a. | 70 cm | ||

b. | 60 cm | ||

c. | 80 cm | ||

d. | 50 cm |

Radius of the cone,

The radius of the cone, height and the slant height form a right triangle.

So, slant height of the cone =

[Apply Pythagorean theorem.]

=

[Substitute the values.]

=

[Substitute the values of 40

=

[Add.]

= 50 cm

[Simplify.]

The slant height of the cone = 50 cm

Correct answer : (4)

5.

The curved surface area of a cone is 2307.90 mm^{2} and the radius of the base of the cone is 21 mm. What is its height? Use π = 3.14 .

a. | 23 mm | ||

b. | 28 mm | ||

c. | 30 mm | ||

d. | 33 mm |

Substitute the

π × 21 ×

[Divide each side by 21.]

π ×

[Simplify.]

3.14 ×

[Substitute π = 3.14.]

[Divide each side by 3.14.]

[Simplify.]

Slant height of the cone = 35 mm

Let

Slant height of the cone =

[Formula.]

35 =

[Substitute the values.]

35

[Squaring on both sides.]

1225 =

[Substitute the values of 35

1225 - 441 =

[Subtract 441 from each side.]

784 =

[Simplify.]

[Square on both the sides.]

28 =

[Simplify.]

The height of the cone = 28 mm

Correct answer : (2)

6.

The total surface area of a cone is 60 square inches. If its slant height is four times the radius, then what is the base diameter of the cone? Use $\pi $ = 3 .

a. | 12 in. | ||

b. | 8 in. | ||

c. | 4 in. | ||

d. | 2 in. |

Slant height

3 ×

[Substitute

12

[Multiply.]

15

[Add.]

[Divide each side by 15.]

[Simplify.]

[Take square root on both sides.]

[Simplify.]

Base radius of the cone = 2 in.

Base diameter of the cone = 2 × radius = 2 × 2 = 4 in.

Correct answer : (3)

7.

A circular cone with a base radius of 13 cm has a surface area of 2122.64 cm^{2}. What would be its slant height?

a. | 36 cm | ||

b. | 39 cm | ||

c. | 26 cm | ||

d. | 52 cm |

π × 13 ×

[Substitute the value of the radius.]

13π

[Multiply.]

13π

[Subtract 169π from each side.]

13π

[Simplify.]

[Divide each side by 13π.]

The slant height of the cone is 39 cm.

Correct answer : (2)

8.

Find the surface area of the square pyramid. [Given $a$ = 12 cm, $b$ = 14 cm.]

a. | 580 cm ^{2} | ||

b. | 480 cm ^{2} | ||

c. | 500 cm ^{2} | ||

d. | 530 cm ^{2} |

=

Area of the lateral face =

[Lateral face is a triangle with base 12 cm and height 14 cm.]

Surface area of the square pyramid = base area + lateral area

[Formula.]

= base area + 4 × area of lateral face

= 12 × 12 + 4× 84

[Substitute the values.]

= 144 + 336

[Multiply.]

= 480

[Add.]

The surface area of the square pyramid is 480 cm

Correct answer : (2)

9.

What is the slant height of the square pyramid if its height ($h$) is 8 cm and its base ($a$) is 12 cm?

a. | 60 cm | ||

b. | 20 cm | ||

c. | 10 cm | ||

d. | 30 cm |

Base of the pyramid is a square.

Length of each side of a square = 12 cm

Slant height of the pyramid = EG

From the figure, EFG is a right triangle.

According to the Pythagoras theorem EG

[Formula.]

FG is parallel to AB and length of FG =

EG

EG

[Substitute the values.]

[Take square root on both sides.]

The slant height of the pyramid with square base = 10 cm

Correct answer : (3)

10.

Christina distributed paper hats to children on a Christmas eve. The hats are in the shape of a cone with base radius of 9 cm and a slant height of 18 cm. Find the surface area of the paper used to make a hat. Use $\pi $ = 3.14 .

a. | 513.98 cm ^{2} | ||

b. | 30.14 cm ^{2} | ||

c. | 508.68 cm ^{2} | ||

d. | 162 cm ^{2} |

Curved surface area of the hat = π

[Formula.]

= π × 9 × 18

[Substitute the values.]

= 162π

[Multiply.]

= 162 × 3.14

[Substitute the value of π = 3.14.]

= 508.68 cm

So, the surface area of the paper used to make a hat = 508.68 cm

Correct answer : (3)