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**Total Surface Area of Cylinder:** A cylinder is a geometric figure formed by two parallel planes intersecting in a straight line. The planes are called the bases of the cylinder, and the line that intersects is called the cylinder’s axis. The space between the two bases is called the cylinder’s volume.

A cylinder’s surface area is the total area of all the surfaces that make up the cylinder. The surface area of a cylinder is made up of the areas of the cylinder’s two bases and the area of its curved surface. The area of a base is the product of its length and width. The area of a curved surface is the product of its height and circumference.

## Surface Area of Cylinder

The surface area of a cylinder is the total area of all the surfaces of the cylinder. The surface area of a cylinder is calculated by multiplying the length of the cylinder by the width of the cylinder by the height of the cylinder.

## Curved Surface Area (CSA) of Cylinder

A cylinder is solid with two parallel circular bases. The curved surface area of a cylinder is the sum of the areas of the two circular bases and the area of the curved surface between them.

### Total Surface Area (TSA) of Cylinder

= π × radius of cylinder × height of the cylinder

= π × 2 cm × 5 cm

= π × 10 cm

= 31.4 cm

### Derivation of the Formula of Surface Area of Cylinder

- The surface area of a cylinder is the sum of the surface areas of the two circular bases and the surface area of the cylindrical body.
- The surface area of a circular base is πr².
- The surface area of a cylindrical body is πrL.
- The surface area of a cylinder is πr²+πrL.

## Solved Examples

Example 1: Find the total surface area of a cylinder with a radius of 7 cm and a height of 14 cm.

Solution:

The formula for the total surface area of a cylinder is given as:

Total surface area = 2πr(r + h)

Where r is the radius of the base of the cylinder, and h is the height of the cylinder.

Substituting the given values, we get:

Total surface area = 2π(7)(7 + 14)

Total surface area = 2π(7)(21)

Total surface area = 294π

Total surface area ≈ 922.98 cm²

Therefore, the total surface area of the cylinder is approximately 922.98 cm².

Example 2: A cylindrical drum has a diameter of 1.2 m and a height of 2.4 m. Calculate the total surface area of the drum.

Solution:

First, we need to find the radius of the base of the cylinder. The diameter is given as 1.2 m, so the radius is half of that, which is 0.6 m.

Using the formula for the total surface area of a cylinder, we get:

Total surface area = 2πr(r + h)

Substituting the given values, we get:

Total surface area = 2π(0.6)(0.6 + 2.4)

Total surface area = 2π(0.6)(3)

Total surface area = 3.6π

Total surface area ≈ 11.31 m²

Therefore, the total surface area of the cylindrical drum is approximately 11.31 m².

Example 3: A cylindrical water tank has a radius of 3 m and a height of 8 m. Calculate the cost of painting the tank if the cost of painting 1 m² is $5.

Solution:

To find the cost of painting the cylindrical tank, we need to first find its total surface area using the formula:

Total surface area = 2πr(r + h)

Substituting the given values, we get:

Total surface area = 2π(3)(3 + 8)

Total surface area = 2π(3)(11)

Total surface area = 66π

Total surface area ≈ 207.35 m²

The cost of painting 1 m² is given as $5, so the cost of painting 207.35 m² is:

Cost = 207.35 × 5

Cost = $1036.75

Therefore, the cost of painting the cylindrical water tank is $1036.75.

## FAQs

Q: What is the Total Surface Area of a Cylinder?

A: The Total Surface Area of a Cylinder is the sum of the areas of its curved surface and two circular bases.

Q: What is the formula for the Total Surface Area of a Cylinder?

A: The formula for the Total Surface Area of a Cylinder is 2πr(h + r), where r is the radius of the base and h is the height of the cylinder.

Q: How do you find the curved surface area of a cylinder?

A: The curved surface area of a cylinder is the product of its height and the circumference of its base, which is 2πr.

Q: How do you find the area of the two circular bases of a cylinder?

A: The area of each circular base of a cylinder is πr^2, where r is the radius of the base.

Q: What is the unit of measurement for Total Surface Area of a Cylinder?

A: The unit of measurement for Total Surface Area of a Cylinder is square units, such as square meters or square centimeters.

Q: Can the height and radius of a cylinder be the same?

A: Yes, the height and radius of a cylinder can be the same. In that case, the cylinder is called a “right circular cylinder.”

Q: What are the applications of the Total Surface Area of a Cylinder in real life?

A: The Total Surface Area of a Cylinder is used in many fields, such as architecture, engineering, and manufacturing, to calculate the amount of material needed to create cylindrical objects like pipes, tanks, and columns.

For more, visit **Area of Hollow Cylinder**