Volume and Surface Area of a Cylinder

In geometry, a cylinder is a three-dimensional solid figure, containing two parallel circular bases joined by a curved surface, situated at a particular distance from the center of the cylinder. For instance, toilet paper rolls, and plastic cold drink cans are examples of cylinders. A cylinder is characterized by two major properties, i.e., surface area and volume. The word cylinder is derived from a Latin (Cylindrus) word, meaning “roll”, “roller”, and “Tumblr”.

Definition of Cylinder

A cylinder is a three-dimensional solid which contains two parallel bases which are held together by a curved surface, held at a fixed distance. The bases of the cylinder are normally circular, similar to a circle. The bases are held together by a line segment, which is referred to as the axis. The distance from this line segment to the outer surface of the cylinder is called the radius. This can be denoted by ‘r’. The perpendicular distance between the bases of the cylinder is called the height of the cylinder, referred to by ‘h’. 

Parts of a Cylinder

Cylinder is considered to be composed of 2 circles + 1 rectangle. The following image depicts the formation of cylinder:

Types of Cylinder

Geometry consists of four types of cylinders, namely, 

• Elliptic Cylinder: A cylinder base forming an ellipse is called an elliptic cylinder.

• Right Circular Cylinder: The right circular cylinder contains the axes of the two parallel lines which are perpendicular to the center of the base.

• Oblique Cylinder: An oblique cylinder is one whose sides lean over the base. In an oblique cylinder, the sides are not perpendicular to the center of the base. The Leaning Tower of Pisa is a real-world example of an oblique cylinder.

• Right Circular Hollow Cylinder or Cylindrical Shell: Also termed as, a cylindrical shell, contains two right circular cylinders bounded by one side on the other. The point of the axis is common to the intersection and is perpendicular to the central base. Since there is some space inside the cylinder, it is hollow from the inside. 

Formulas of Cylinder

The cylinder is associated with three formulae, finding its applications with area and volume: 

• Lateral Surface Area or Curved Surface Area
• Total Surface Area
• Volume of the Cylinder

Lateral Surface Area or Curved Surface Area of Cylinder

Curved surface area is also termed a lateral surface area. The area formed by the curved surface of the cylinder i.e. space occupied between the two parallel circular bases is known as CSA. The formula for CSA is given as:

Curved Surface Area (CSA) = 2πrh square units

where,

• ‘h’ is the height
• ‘r’ is the radius

Total Surface Area of Cylinder

So, in order to find out the total surface area of a cylinder, we calculate the curved surface area and the area of two circles.

Total surface area of the cylinder is defined as the total area occupied by it. A cylinder consists of two circles along with a curved sheet. The total surface area of a cylinder can be calculated by the combination of curved surface area and the area of two circles. 

Curved Surface Area(CSA) = Circumference of the Circle × Height

C.S.A = 2r × h

Area of a Circle = πr²

Total Surface Area (TSA) = Curved Surface Area + 2(Area of a circle)

We know, 

Curved Surface Area = 2πrh 

Area of circle = πr²

Total Surface Area (T.S.A) = 2πrh + 2πr² = 2πr(h+r) square units.

where,

• h is Height
• r is Radius of Cylinder

Volume of Cylinder

Volume of cylinder is referred to as the density or amount of space it occupies.

We have, 

Volume of a Cylinder = Area of a Circle × Height

Since, we have an area of a circle = πr²

Volume  = πr² × h

where,

• h is Height
• r is Radius of Cylinder

Sample Problems on Volume and Surface Area of a Cylinder

Problem 1: Compute the total surface area of the cylinder, with a radius of 5cm and height of 10cm?

Solution:

Since, we know, 

Total surface area of a cylinder, A = 2πr(r+h) square units

Therefore, A = 2π × 5(5 + 10) = 2π × 5(15) 

= 2π × 75 = 150 × 3.14 

= 471 cm²

Problem 2: What is the volume of a cylindrical shape water container, that has a height of 7cm and diameter of 10cm?

Solution: 

Given,

Diameter of the container = 10cm

Thus, radius of the container = 10/2 = 5cm

Height of container = 7cm

As we know, from the formula,

Volume of a cylinder = πr²h cubic units.

Therefore, volume of given container, V = π × 52 × 7

V = π × 25 × 7 = (22/7) × 25 × 7 = 22 × 25

V = 550 cm³

Problem 3: Alex wants to purchase a cylindrical can with a radius equivalent to 5 inches. The can contains 1 gallon of oil. Find the height of the cylinder. 

Solution:

Volume V is given by= 1 gallon

1 gallon= 231 cubic inches

Radius r = 5 inches

Volume f the cylinder is given by, 

 V = πr²h

231 = 22/7 × (5)² × h

(231 × 7)/(22 × 25) = h

h = 2.94 inches.

Therefore, the height is equivalent to 2.94 inches.

Problem 4: A water tank has a radius of 40 inches and a height of 150 inches. Find the area. 

Solution:

Water tank is cylindrical in nature. 

Total Surface Area of a cylinder is given by,  2πr(h+r)

TSA = 2 × 22/7 × 40(150 + 40)

TSA = 2 × 22/7 × 40 × 190

TSA = 440/7 × 7600

TSA = 3344000/ 7

Area = 47,7142.857 sq.inches.

Problem 5: Find the volume of the cylinder having a radius of 5 units and a height of 8 units?

Solution:

We have, 

Radius,r = 5 units

Height,h = 8 units

Volume of the cylinder, V = πr²h cubic units.

V = (22/7) × 5² × 8

V = 22/7 × 25 × 8

V= 628.57 Cubic units.

Hence, the volume of the cylinder is 628.57 cubic units.

FAQs on Volume and Surface Area of a Cylinder

What is the surface area of the cylinder?

Surface area of cylinder is defined as the area of all the surfaces of the cylinder and found by adding all the surfaces of the cylinder.

What is the total surface area volume of a cylinder?

• Total surface area of the cylinder is the sum of all the surfaces of the cylinder.
• Whereas volume of cylinder is defined as the capacity of the cylinder, it is defined as the total space occupied by the cylinder.

What is the formula for volume and surface area of cylinder?

Formula for surface area of cylinder is,

A = 2πr(h+r) square units.

Formula for volume of cylinder is,

V = πr²h cubic units.