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Surface Area of a Cylinder(Definition, Formulas & Examples)

surface area of a cylinder

The surface area of 3D shapes refers to a measure of the total area that the surface of the objects like cubes, cuboids, spheres, cones, prisms, and pyramids of all the surfaces occupies. It is in square units, like $m^{2}$, $cm^{2}$, $in^{2}$, $ft^{2}$, etc. The total surface area of a cylinder can be calculated if we calculate the area of the two bases and the area of the curved face.

Let’s learn how to find the surface area of a cylinder and its formulas.

Cylinder – A 3D Solid Shape

A cylinder is one of the basic 3D shapes, in geometry, which has two parallel circular bases at some distance. The two circular bases are joined by a curved surface, at a fixed distance from the centre. The line segment joining the centre of two circular bases is the axis of the cylinder. The distance between the two circular bases is called the height of the cylinder. LPG gas cylinder is one of the real-life examples of cylinders. 

Since the cylinder is a three-dimensional shape, therefore it has two major properties, i.e., surface area and volume. The total surface area of the cylinder is equal to the sum of its curved surface area and the area of the two circular bases. The space occupied by a cylinder in three dimensions is called its volume.

Some of the important properties of the cylinder are as follows:

  • The bases of the cylinder are always congruent and parallel to each other.
  • If the axis of the cylinder is at a right angle to the base and the bases are exactly over each other, then it is called a “Right Cylinder”.
  • If one of the bases of the cylinder is displayed sideways, and the axis does not produce the right angle to the bases, then it is called an “Oblique Cylinder”.
  • If the bases are circular, then it is called a right circular cylinder.
  • The best alternative to the circular base of a cylinder is an ellipse. If the base of the cylinder is elliptical in shape, then it is called an “Elliptical Cylinder”.
  • If the locus of a line moves parallel and fixed distance from the axis, a circular cylinder is produced.
  • A cylinder is similar to a prism since it has the same cross-section everywhere.
surface area of a cylinder

For ease of understanding, the right cylinder or right circular cylinder is considered for studying the different properties of the cylinder.

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Right Circular Cylinder

A cylinder whose bases are circular in shape and parallel to each other is called the right circular cylinder. It is a three-dimensional shape. The axis of the cylinder joins the center of the two bases of the cylinder. This is the most common type of cylinder used in day-to-day life. It is different from the oblique cylinder which does not have parallel bases and resembles a tilted structure.

surface area of a cylinder

Parts of Right Circular Cylinder

The three parts of the right circular cylinder are:

  • Top Circular Base
  • Curved Lateral Face
  • Bottom Circular Base

Properties of Right Circular Cylinder

These are the properties of a right circular cylinder.

  • The line joining the centers of the circle is called the axis.
  • When we revolve a rectangle about one side as the axis of revolution, a right cylinder is formed.
  • The section obtained on cutting a right circular cylinder by a plane contains two elements and the parallels to the axis of the cylinder is the rectangle.
  • If a plane cuts the right cylinder horizontally parallel to the bases, then it’s a circle.

What is the Surface Area of a Cylinder?

The surface area of a cylinder is the sum of the area of the circular bases($2$ bases) and the area of the curved face($1$ curved face) of the cylinder. 

The surface area of a cylinder is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of a cylinder

  • Curved Surface Area
  • Total Surface Area

Curved Surface Area of a Cylinder

The curved surface area of a cylinder is the surface area covered by its curved surface only. The formula for calculating the curved surface area of the cylinder is $2 \pi r h$, where $r$ is the radius of the base of the cylinder and $h$ is the height of the cylinder and $\pi$ is a mathematical constant, and is taken as $\frac {22}{7}$ or $3.14$ during calculation.

Derivation of Curved Surface Area of a Cylinder Formula

A right circular cylinder can be considered as a $3D$ shape formed by keeping a number of circles one over the other.

surface area of a cylinder

Let’s consider a cylinder of base radius $r$ and height $h$ formed by a heap(or stack) of height $h$ formed by circles of radius $r$.

The curved surface area of this cylinder will be the sum of the circumferences of all these circles.

Circumference of a circle of radius $r$ is $2 \pi r$.

Therefore, the sum of the circumference of all circles will be $2 \pi r + 2 \pi r + 2 \pi r + … + \left( h \text{ times} \right) = 2 \pi r \times h = 2 \pi rh$.

The formula for the curved surface area of a cylinder is $2 \pi rh$.

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Total Surface Area of a Cylinder

The total surface area of the cylinder is obtained by adding the area of the two bases and the area of the curved surface. Thus, the formula for the total surface area of the cylinder is given as:

The total surface area of the cylinder = (Area of two bases) + (Area of the curved surface). Since the bases of the cylinder are circular in shape, their combined area will be $\pi r^{2} + \pi r^{2} = 2 \pi r^{2}$. 

We already know that the curved surface area of a cylinder is $2 \pi rh$.

Therefore, the total surface area of cylinder is $2 \pi r^{2} + 2 \pi rh = 2 \pi r \left(r + h \right)$.

The formula for the total surface area of a cylinder is $2 \pi r \left(r + h \right)$.

Examples

Ex 1: The diameter of the base of a cylinder is $12 cm$ and the height is $8 cm$. Find the total surface area of the solid cylinder.

Diameter of base of cylinder  $d = 12 cm$

Radius of base of cylinder  $r = \frac {d}{2} = \frac {12}{2} = 6 cm$

Height of cylinder $h = 8 cm$

Total surface area = $2 \pi r \left(r + h \right) = 2 \times \frac {22}{7} \times 6 \left(6 + 8 \right) = 528 cm^{2}$

Ex 2: A cylindrical pillar is $50 cm$ in diameter and $7 m$ in height. Find the cost of painting the curved surface of the pillar at the rate of ₹$12$ per sq. m.

Diameter of cylindrical pillar  $d = 50 cm$

Radius of cylindrical pillar $r = \frac {d}{2} = \frac {50}{2} = 25 cm = 0.25 m$

Height of cylindrical pillar $h = 7 m$

The area to be painted on a cylindrical pillar is equal to the lateral surface area of the cylindrical pillar.

Curved surface area of cylinder = $2 \pi rh$

Area to be painted on one pillar = $2 \times \frac {22}{7} \times 0.25 \times 7 = 11 m^{2}$.

Rate of painting = ₹$12$ per sq. m.

Therefore, cost of painting the pillar = $11 \times 12 =$ ₹ $132$.

Ex 3: The curved surface area of a right circular cylinder of base radius $7 cm$ is $110 cm^{2}$. Find the height of the cylinder.

Radius of base of the cylinder $r = 7 cm$

Curved surface area of the cylinder = $110 cm^{2}$

Let the height of the cylinder be $h$

Curved surface area of cylinder = $2 \pi rh$

Therefore, $2 \times \frac {22}{7} \times 7 \times h = 110 =>44h = 110 => h = \frac {110}{44} = 2.5 cm$

Ex 4: How many square meters of metal sheet is required to make a closed cylindrical tank of height $1.8 m$ and base diameter $140 cm$?

Diameter of cylindrical tank  $d = 140 cm$

Radius of cylindrical tank $r = \frac {d}{2} = \frac {140}{2} = 70 cm = 0.7 m$

Height of cylindrical tank $h = 1.8 m$

The amount of metal sheet required is the total surface area of the cylinder.

Total surface area of cylinder = $2 \pi r \left(r + h \right) = 2 \times \frac {22}{7} \times 0.7 \times \left(0.7 + 1.8 \right) = 11 m^{2}$.

Therefore, the metal sheet required to make a closed cylindrical tank is $11 m^{2}$.

Conclusion

The surface area of a 3D shape (solid object) is a measure of the total area that the surface of the object occupies. There are two types of surface areas in a cylinder, viz., curved surface area calculated using the formula $2 \pi rh$ and total surface area calculated using the formula $2 \pi r \left(r + h \right)$, where $r$ and $h$ are the radius of the circular base and height of a cylinder respectively.

Practice Problems

  1. Find the total surface area of a cylinder whose radius is $5 cm$ and height is $7 cm$.
  2. The cost of painting a cylindrical container is ₹$0.40 per $cm^{2}$. Find the cost of painting $20$ containers of radius $50 cm$, and height $80 cm$.
  3. Find the height of a cylinder if its total surface area is $2552 in^{2}$ and the radius is $14 in$.
  4. Find the curved surface area of a cylinder whose diameter is $56 cm$ and height is $20 cm$.
  5. The curved surface area of a cylinder is $144 ft^{2}$. If the radius of the cylinder is $7 ft$, find the height of the cylinder.
  6. Which of the following cylinder has the largest surface area?
    • A cylinder with a radius of $10 ft$ and a height of $20 ft$.
    • A cylinder with a radius of $20 ft$ and a height of $10 ft$.
    • A cylinder with a diameter of $30 ft$ and a height of $20 ft$.
  7. What is the total surface area of a cylinder with a radius of $12$ inches and a height of $14$ inches? $\left( \pi = 3.14 \right)$.
  8. A cylinder has a total surface area of $484$ squared feet. If the cylinder’s height is $4$ feet, what is the radius of the cylinder? $\left( \pi = \frac {22}{7} \right)$ 
  9. The cost of painting a cylindrical container is ₹$6$ per square inch. How much would it cost to paint $10$ cylindrical containers with a radius of $10$ inches and a height of $20$ inch? Use $ \pi = 3.14$

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FAQs

How do you find a surface area of a cylinder?

There are two types of surface area in a cylinder.
a) Curved Surface Area: The curved surface area of a cylinder is the surface area covered by its curved surface only. The formula for calculating the curved surface area of the cylinder is $2 \pi r h$. 
b) Total Surface Area: The total surface area of the cylinder is obtained by adding the area of the two bases and the area of the curved surface. The formula for calculating the total surface area of the cylinder is $2 \pi r \left(r + h \right)$.

where $r$ is the radius of the base of the cylinder and $h$ is the height of the cylinder and $\pi$ is a mathematical constant and is taken as $\frac {22}{7}$ or $3.14$ during calculation.

What is the formula for the surface area of a cylinder?

a) The formula for calculating the curved surface area of the cylinder is $2 \pi r h$.
b) The formula for calculating the total surface area of the cylinder is $2 \pi r \left(r + h \right)$.

What are the CSA and TSA of a cylinder?

A cylinder consists of two types of surface, one is a curved surface and the other is a circular basis. The area of both circular bases is equal. The surface area of the cylinder can be classified into two types
a) Curved surface area (CSA) 
b) Total surface area (TSA)

Are CSA and LSA the same?

a) Curved Surface Area (CSA): It includes the area of all the curved surfaces. 
b) Lateral Surface Area (LSA): It includes the area of all the surfaces excluding the top and bottom areas. 
c) Total Surface Area (TSA): It includes the area of all the surfaces of the object including the bases.

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