# Surface Area of Pyramids and Cones Worksheet - Page 4

Surface Area of Pyramids and Cones Worksheet
• Page 4
31.
What is the slant height of the pyramid, if $h$ is the height of a pyramid standing on a square base of side $a$ units?

 a. √ (4 $h$2 + $a$2)/2 units b. √ (2 $h$2 + $a$2)/2 units c. (2 $h$2 + $a$2) units d. √ (4 $h$2 + 2 $a$2)/2 units

#### Solution:

Height of the pyramid = h units

Base of the pyramid is a square.

Length of each side of a square = a units.

Slant height of the pyramid = EG

From the figure, EFG is a right triangle.

According to the Pythagoras theorem EG2 = EF2 + FG2
[Formula.]

FG is parallel to AB and length of FG = AB / 2 = a / 2 units.

EG2 = EF2 + FG2

EG2 = h2 + ( a / 2)2
[Substitute the values.]

EG2 = h2 + a2/4
[Simplify (a/22).]

EG2 = (4 h2 + a2)/4

√EG2 = √ ((4 h2 + a2)/4)
[Take square root on both sides.]

EG = √ (4h2 + a2)/2
[Simplify.]

The slant height of the pyramid with a square base = √ (4 h2 + a2)/2 units.

32.
Perpendicular distance from the base to the opposite vertex is called __________.
 a. Base b. Either height or slant height c. Slant height d. Height

#### Solution:

The perpendicular distance from the base to the opposite vertex is called the height.

33.
Perpendicular distance from the edge of the base to the opposite vertex of a pyramid is called _________________.
 a. Slant height b. Height c. Base d. None of the above

#### Solution:

Perpendicular distance from the edge of the base to the opposite vertex is called the slant height.

34.
Laura distributed paper hats to children on the Christmas eve. The hats are in the shape of a cone with a base radius of 6 cm. and a slant height of 17 cm. Find the surface area of the paper used to make each hat. [Use π = 3.14.]
 a. 102 cm2 b. 160.14 cm2 c. 320.28 cm2 d. 336.58 cm2

#### Solution:

Base radius of the hat, which is in the shape of cone, r = 6 cm. and its slant height, l = 17 cm.

Curved surface area of the hat = πrl
[Formula.]

= π × 6 × 17
[Substitute the values.]

= 102π
[Multiply.]

= 102 × 3.14
[Substitute the value of π = 3.14.]

= 320.28 cm.2

So, the surface area of the paper used to make a hat = 320.28 cm.2

35.
A circular cone is 6 in. high. The radius of the base is 8 in. What is the curved surface area of the cone?
 a. 251.2 in.2 b. 257.2 in.2 c. 259.8 in.2 d. 245.2 in.2

#### Solution:

Height of a circular cone, h = 6 in. and its base radius, r = 8 in.

Slant height of the cone = h2 + r2
[Formula.]

= 62 + 82
[Substitute the values.]

= (36 + 64)
[Substitute the values of 62 and 82.]

= 100

= 10
[Simplify.]

Slant height of the cone = 10 in.

Curved surface area of the cone = πrl
[Formula.]

= 3.14 × 8 × 10
[Substitute the values.]

= 251.2 in.2

The curved surface area of the cone is 251.2 in.2.

36.
The diameter of an ice cream cone is 4 cm. and the slant height is 9 cm. What is the surface area of the cone? [Use π = 3.14.]
 a. 65.08 cm2 b. 75.48 cm2 c. 69.08 cm2 d. 73.08 cm2

#### Solution:

Diameter of an ice cream cone = 4 cm. and its slant height, l = 9 cm.

Radius of an ice cream cone, r = diameter / 2 = 4 / 2 = 2 cm.
[Substitute diameter = 4.]

Surface area of the cone = πrl + πr2
[Formula.]

= π × 2 × 9 + π × 22
[Substitute the values.]

= 18π + 4π

= 22π

= 22 × 3.14
[Substitute the value of π = 3.14.]

= 69.08
[Multiply.]

The surface area of the cone is 69.08 cm.2.

37.
The circumference of the base of a conical tent is 25.12 m. and its slant height is 8 m. Find the area of the canvas used in making the tent.[Use $\pi$ = 3.14.]
 a. 102.48 m2 b. 106.48 m2 c. 98.48 m2 d. 100.48 m2

#### Solution:

Area of canvas required = lateral area of the conical tent

Circumference of the conical tent = 2πr = 25.12 m.

2 x 3.14 x r = 25.12
[Substitute the value of π = 3.14.]

6.28r = 25.12
[Multiply 2 with 3.14.]

6.28r6.28 = 25.126.28
[Divide each side by 6.28.]

r = 4
[Simplify.]

Lateral area of the tent = πrl
[Formula.]

= 3.14 x 4 x 8
[Substitute the values.]

= 100.48 m.2
[Multiply.]

So, area of the canvas required is 100.48 m.2.

38.
What is the slant height of the cone, if its lateral area is 720 ft2 and its radius is 12 ft? (Round the answer to the nearest hundredth) [Use π = 3.14.]

 a. 17.11 ft b. 19.11 ft c. 21.11 ft d. None of the above

#### Solution:

Lateral area of a cone = πrl = 720 ft2

Radius of the cone = 12 ft

π x 12 x l = 720
[Substitute the value of r in step 1.]

(π x 12 x l)/12 = 72012
[Divide each side by 12.]

πl = 60
[Simplify.]

3.14 x l = 60
[Substitute π = 3.14.]

(3.14 x l)/3.14 = 603.14
[Divide each side 3.14.]

l = 19.108 19.11
[Simplify.]

The slant height of the cone is 19.11 ft.

39.
The height of a cone is 16 in. and its diameter is 24 in. What is the slant height of the cone?

 a. 22 in. b. 16 in. c. 18 in. d. 20 in.

#### Solution:

Height of a cone, h = 16 in. and its diameter = 24 in.

Radius of the cone, r = diameter / 2 = 24 / 2 = 12 in.

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

So, slant height of the cone = √(h2 + r2)
[Apply Pythagorean theorem.]

= √(162 + 122)
[Substitute the values.]

= √(256 + 144)
[Substitute the values of 162, 122.]

= √400

= 20 in.
[Simplify.]

The slant height of the cone = 20 in.

40.
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. 30 mm b. 33 mm c. 28 mm d. 23 mm

#### Solution:

The curved surface area of a cone = πrl = 2307.90 mm.2 and its radius is 21 mm.

Substitute the r value in πrl = 2307.90 mm.2

π x 21 x l = 2307.90

(π x 21 x l)/21 = 2307.9021
[Divide each side by 21.]

π x l = 109.90
[Simplify.]

3.14 x l = 109.90
[Substitute π = 3.14.]

(3.14 x l)/3.14 = 109.903.14
[Divide each side by 3.14.]

l = 35 mm
[Simplify.]

Slant height of the cone = 35 mm.

Let h be the height of the cone.

Slant height of the cone = √(h2 + r2)
[Formula.]

35 = √(h2 + 212)
[Substitute the values.]

352 = (√(h2 + 212))2
[Squaring both sides.]

1225 = h2 + 441
[Substitute the values of 352 and 212.]

1225 - 441 = h2 + 441 - 441
[Subtract 441 from each side.]

784 = h2
[Simplify.]

√784 = √h2
[Square on both the sides.]

28 = h2
[Simplify.]

The height of the cone = 28 mm.