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Volume of a Cone



r: the radius of the base
h: height

The volume of a cone is given by:
V =
 π ∙ r2 ∙ h / 3,
where π
 ∙ r2 is the base area of the cone.

π defines the ratio of any circle's circumference to its diameter and is approximately equal to 3.141593, however the value 3.14 is often used.

Example 1: Find the volume of cone, whose radius is 8 cm and height is 5 cm.
Solution:

The formula to find the volume of cone is given by:

V = π ∙ r2 ∙ h / 3
V =
 π ∙ 82 ∙ 5 / 3
V =
 1004.8 / 3 = 334.93 cm3
 
Thus, the volume of the cone is 334.93 cm3.


Example 2: Find the volume of a cone whose base radius is 2.1 cm and height is 6 cm using π = 22/7
Solution:

The volume of a cone is defined as:
V = π ∙ r2 ∙ h / 3
V =
 22/7 ∙ 2.12 ∙ 6 / 3
V =
 27.72 cm3
So, the volume of the cone is 27.72 cm3
 

Example 3: Find the volume of a cone, whose diameter is 8 cm and the height is 11 cm.
Solution

As in previous examples:
V = π ∙ r2 ∙ h / 3

r is the determine by D/2, which is: 8cm / 2 = 4cm, insert the value of r we have:
V = π ∙ 42 ∙ 11 / 3
V = 552.64/3 cm3
V = 184.21 cm3
 
Thus the volume of the cone is 184.21 cm3.


Example 4: Find the height of a right circular cone whose volume is 169 cm3 and radius is 4cm
Solution:
V = 1/3
π ∙ r2 ∙ h
V = 169 = 1/3
 ∙ 3.14 ∙ 42 ∙ h
V = 169 = 16.75
 ∙ h
V = h = 169/16.75
V = h = 10.09 cm
The height of the right circular cone is 10.09 cm.


Example 5: Find the height of a cone whose volume is 22 cm3 and the radius of the cone = 1cm, use π = 22/7
Solution
V = 1/3
 ∙ π ∙ r2 ∙ h = 22
V = 1/3
 ∙ 22/7 ∙ 1 ∙ 1h = 22
V = h = (22
∙ 7 ∙ 3)/22
V = h = 21 cm.

Example 6: The circumference of the base of a 9 m high conical tent is 44 m. find the volume of the air contained in it, use π = 22/7
Solution:
Circumference of the base:
P = 2
 ∙ π ∙ r = 44m
r = P / ( 2
 ∙ π )
r = 44 / ( 2
 ∙ π ) = 7 m

Height of the conical tent = 9m
Volume of air = 1/3
 ∙ π ∙ r2 ∙ h = 1/3 ∙22/7 ∙ 7 ∙ 7 ∙ 9 = 462 m3


Example 7: The vertical height of a conical tent is 42 dm and the diameter of its base is 5.4 m.
Determine the number of persons it can accommodate if each person uses 2916 dm3 of space use π = 22/7
Solution:
Height: h = 42 dm
Diameter: = 5.4 m = 54 dm  
Radius: r = D/2 = 27 dm 
Volume = 1/3
 ∙ π r2 ∙ h
Volume = 1/3
 ∙ 22/7 ∙ 27 ∙ 27 ∙ 42 = 32076 dm3


Space allowed for 1 person =2916 dm3
Number of persons = 32076 / 2916 = 11 Persons.

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