Volume of Right Circular Cylinder

Volume of a right circular cylinder is the space occupied by it. A cylinder is a geometric three-dimensional object that is very common in our daily life such as a coke can, paper roll, etc. It has two parallel and congruent circular bases that are connected by a curved surface (which is formed by the line segments joining the respective points of the bases). There are two types of cylinders:

• Right circular cylinder
• Oblique cylinder

We will study more about the right circular cylinder and its volume in detail here with the help of a few solved examples and practice questions.

A right circular cylinder is a cylinder that has two congruent and parallel circular bases where each line segment that is a part of the lateral or the curved surface is perpendicular to the bases. The volume of a right circular cylinder is the number of unit cubes that can fit into it. The unit of volume is "cubic units". It is expressed as m³, cm³, km³, etc depending upon the given units. Let us see how to find the formula of the volume of a right circular cylinder.

Let us consider a cylinder of radius 'r' and height 'h'. Let us assume that its volume is 'V'. Since the bases of the right circular cylinder are congruent and parallel circles, it is referred to as a circular prism (but in fact, a cylinder is not a prism). We know that the volume of any prism is its base area multiplied by its height.

We know that each base of a right circular cylinder is a circle and hence its area is, πr². The height of the cylinder is 'h'. So its volume is

V = base area × height = πr² × h = πr²h.

By the above formula, we can say that the volume of a right circular cylinder is directly proportional to the square of its radius and also its height which means:

• If the radius of the base becomes double, then the volume becomes four times.
• If the height becomes double, then the volume also becomes double.

As we learned in the previous section, the volume of a right circular cylinder of radius r and height h is V = πr²h. Thus, we follow the below steps to find the volume of the right circular cylinder.

• Step 1: Identify the radius and name it r; Identify its height and name it h.
• Step 2: Find the volume using the formula V = πr²h.
• Step 3: Represent the final answer with cubic units.

Example: Find the volume of a cylinder of radius 3 units and height 7 units. Use π = 3.14.

Solution:

The radius of the cylinder is, r = 3 units.

Its height is, h = 7 units.

Its volume is calculated as: V = πr²h
⇒ V = (3.14) (3)² (7)
⇒ V = 197.82 cubic units.

Therefore, the volume of the given cylinder is 197.82 cubic units.

For a right circular cylinder, if its height (h) and base radius (R) is given, then its volume using integration can be given as shown below.

Consider a volume element as shown in the figure at right, which is at distance z from the center of the cylinder and has thickness dz. The volume of this element is equal to its base area times the thickness.

Volume of element (dV) = Area of base of the element × Thickness = (πR²) dz

Summing up the volume of all such elements will give us the volume of a cylinder. Therefore, the volume of a cylinder is equal to,

Volume of the cylinder = Sum of all volumes of all such elements = Definite Integral of the volume of this element from z=0 to z=h

Volume of a Right Circular Cylinder, V = \(int_{z = 0}^{h} \pi R^2 \,dz = \pi R^2 \bigg[ z \bigg]_0^h = \pi R^2 \bigg[ h - 0 \bigg] = \pi R^2h \)

Note: Volume of an oblique cylinder is same as that of volume of a right cylinder, but vertical height is taken instead of the cylinder's slant height.

How Do You Find the Volume of a Circular Cylinder?

The volume (V) of a circular cylinder of radius r and height h is calculated using the formula V = π r²h.

What Is the Volume of a Right Circular Cylinder of Base Radius 7 cm and Height 10 cm?

The radius of the cylinder is, r = 7 cm. Its height is, h = 10 cm. Its volume is, V = πr²h
⇒V = (3.14) (7)² (10)
⇒V = 1538.6 cm³.

What Is the Difference Between the Cylinder and Right Circular Cylinder?

A cylinder can be either an oblique or a right circular cylinder. In a right circular cylinder, the bases are parallel and congruent circles where each line segment of the lateral curved surface is perpendicular to the bases. An oblique cylinder is a cylinder that is not a right circular cylinder.

What Is the Formula for Finding the Volume of a Right Circular Cylinder?

The formula to find the volume of a right circular cylinder whose radius is r and height is h is, V = πr²h. i.e., the volume directly varies with the square of the radius, and also it directly varies with the height of the cylinder.

How Do You Find the Volume of a Cylinder if Only Base Area and Height Is Given?

The volume of a cylinder is the capacity of the cylinder which signifies the amount of any material it can hold or the amount of any material that can be immersed in it. From the definition, the volume of the cylinder = Base Area × Height

What Is the Unit Used to Express Volume of a Cylinder?

The volume of a cylinder is expressed in cubic units, like in³, cm³, m³, ft³, etc.