If you are a student of math or engineering, you may have come across the task of finding the total surface area of a cylinder. It may seem like a daunting task, but with the right information and tools, it can be done easily. This article will guide you through the process of calculating the total surface area of a cylinder.

First, let us understand what a cylinder is. A cylinder is a three-dimensional shape that has two parallel bases connected by a curved surface. A cylinder can be either right or oblique. In a right cylinder, the axis is perpendicular to the bases, while in an oblique cylinder, the axis is at an angle to the bases. In this article, we will focus on the right cylinder.

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

2πr(r+h)

Where r is the radius of the circular base and h is the height of the cylinder. π (pi) is a constant that is approximately equal to 3.14.

Let’s break down the formula and understand each component. The first term, 2πr, represents the area of the two circular bases of the cylinder. The second term, 2πrh, represents the area of the curved surface of the cylinder.

To simplify the calculation, we can calculate the area of the circular base separately and then multiply it by two. The area of the circular base is given by the formula:

πr^2

Where r is the radius of the circular base. Therefore, the area of both the circular bases can be calculated by multiplying the area of one circular base by two. This gives us:

2πr^2

Next, we need to calculate the area of the curved surface of the cylinder. We can do this by calculating the lateral surface area of the cylinder. The lateral surface area is the area of the curved surface of the cylinder without the area of the bases. The formula for the lateral surface area of a cylinder is:

2πrh

Where r is the radius of the base and h is the height of the cylinder. Therefore, the total surface area of the right cylinder is given by the formula:

2πr^2 + 2πrh

Now that we understand the formula, let’s take an example to illustrate how to calculate the total surface area of a cylinder.

Example: Find the total surface area of a right cylinder with radius 3 cm and height 6 cm.

Solution:

We can use the formula 2πr^2 + 2πrh to find the total surface area of the cylinder. Let’s substitute the given values into the formula.

Total surface area = 2πr^2 + 2πrh
= 2π(3)^2 + 2π(3)(6) (Substituting r = 3 and h = 6)
= 2π(9) + 2π(18)
= 56.55 cm^2 (Approximately)

Therefore, the total surface area of the given cylinder is 56.55 cm^2.

In conclusion, finding the total surface area of a cylinder is not as difficult as it may seem. By using the correct formula and following a step-by-step process, you can easily find the total surface area of any cylinder.

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