What is a Cylinder?
A cylinder is a three dimensional solid object. It consists of two congruent plane circular surfaces called bases of the cylinder and one curved lateral surface. There are different types of cylinders: oblique, elliptical, rectangular, and right circular.
However, we usually refer to the right-circular cylinder using the term “cylinder”. A cylinder is called a right–circular cylinder when one of its bases lies exactly above the other base. If one base is not exactly above the other base, then the cylinder is known as an “oblique cylinder”.
A cylinder is a 3D-shaped object that can also be obtained by rotating a rectangle along one of its sides.
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What is the Volume of a Cylinder?
In math, volume is the region or space occupied by a three-dimensional (or 3D) cylinder. As volume is a 3D quantity, the volume of a cylinder is expressed in cubic units like cubic feet, cubic meters, cubic inches, etc.
To find the volume of a cylinder, we need two parameters:
1. Radius of the cylinder
2. Height of the cylinder
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The Volume of a Cylinder formula
Let \( V \) be the volume of a cylinder, \( R \) be the base radius and \( h \) be its height.
Then,
Volume of the cylinder is given by,
\( V~=~\pi R^2h \) (cubic units)
(i) For a right-circular cylinder,
\( V~=~\pi R^2h \) (cubic units)
(ii) For a hollow cylinder,
\( V~=~\pi (R^2~-~r^2)h \) (cubic units)
Here, \( (R^2-r^2) \) is the thickness or corresponding radius of the cylinder.
(\( R \), is the outer radius and \(r\) is the inner radius.)
And \(‘h’\) is the height of the cylinder.
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Volume of a Cylinder Examples
Example 1: The radius of a cylinder is 8 inches and the height of the cylinder is 12 inches. Find the volume of the cylinder.
(Assume that the value of \( \pi \) as 3.14)
Solution:
Given,
Radius of the cylinder (r) = 8 inches
Height of the cylinder (h) = 12 inches
\( \pi r^2h \) [Surface area of cylinder formula]
\( =~ 3.14~\times ~8^2~\times ~12 \) [Replace r with 8 and h with 12]
\( = 3.14~\times ~64~\times ~12 \) [Simplify]
\( \approx 2411.52 \) [Simplify]
The volume of the cylinder is 2411.52 cubic inches.
Example 2: The radius of a cylinder is 5 feet and the height of the cylinder is 10 feet. Find the volume of this cylinder.
(Assume the the value of \( \pi \) to be 3.14)
Solution:
Given,
Radius of the cylinder “r” = 5 inches
Height of the cylinder “h” = 10 inches
\( \pi r^2h \) [Surface area of cylinder formula]
\( = 3.14~\times~ 5^2~\times~ 10 \) [Replace r with 8 and h with 12]
\( = 3.14~\times~ 25~\times~ 10 \) [Simplify]
\( \approx 785 \) [Simplify]
The volume of the cylinder is 785 cubic feet.
Example 3: Robert has a cylindrical water tank. The radius of the tank is 50 inches and the height of the tank is 1000 inches. The cost of 100 cubic inches of water is $0.50. Find the total cost of water that Robert’s tank can hold.
Solution:
The cost of 1000 cubic inches of water is $0.50. So, first we calculate the capacity of the water tank in terms of cubic inches. Then we divide the volume of the tank by 1000 cubic inches to find the total cost.
Radius of the cylinder “r” = 50 inches
Height of the cylinder “h” = 100 inches
\( \pi r^2h \) [Surface area of cylinder formula]
\( = 3.14~\times~ 50^2~\times~ 100\) [Replace r with 8 and h with 12]
\( = 3.14~\times ~2500~\times~ 100\) [Simplify]
\( \approx 7850000 \) [Simplify]
The volume of the cylinder is 785,000 cubic inches.
Therefore the total cost of water that the tank can hold is
\( \frac{785000}{1000} ~= ~785 \)
Therefore the total cost of one tank of water is $785.
Example 4: The height and diameter of an oxygen cylinder are 20 inches and 100 inches, respectively. Find the quantity of oxygen that this cylinder can contain in cubic inches.
Solution:
Radius of the cylinder (r) = 20 inches
Height of the cylinder (h) = 100 inches
\( \pi r^2h \) [Surface area of cylinder formula]
\( = 3.14~\times ~20^2~\times~ 100 \) [Replace r with 8 and h with 12]
\( = 3.14~\times~ 400~\times~ 100 \) [Simplify]
\( \approx 125600 \) [Simplify]
The oxygen cylinder can hold 125600 cubic inches of oxygen.
The base of the cylinder is a circle. The circumference of the cylinder can be calculated using the formula:
Circumference of cylinder formula = 2πr
An oblique cylinder is a cylinder that is tilted about its base. The lateral surface of the cylinder is not perpendicular to the base. One of the circular bases of the cylinder does not lie above the other.
Steps:
(i) Volume and height of the cylinder should be in the same units .
(ii) Divide the volume by radius squared and pi to get the required height of the cylinder in the same unit as that of radius.
(i) Volume and radius of the cylinder should be in the same units .
(ii) Divide the volume by pi and the height
(iii) Take the square root of the result from step (ii) to get the required radius