**Introduction**

The concept of Surface Area is a fundamental aspect of geometry, an important branch of mathematics. One of the most common objects students learn about surface area is the cylinder. It’s a three-dimensional figure with two parallel, congruent bases. Understanding how to calculate the surface area of a cylinder is crucial not only in academic settings but also in real-world applications.

**Grade Appropriateness**

Learning the surface area of a cylinder is typically appropriate for students in the 7th to 9th grade, coinciding with the introduction of more advanced geometry concepts in the middle school curriculum. However, the idea can also be revisited in higher grades for more complex applications.

**Math Domain**

This topic falls under Geometry, which deals with sizes, shapes, and properties of figures and spaces. Specifically, it pertains to the sub-domain of measurement and dimensional analysis.

**Applicable Common Core Standards**

The relevant Common Core Standards for this topic are:

*7.G.B.6:* “Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.”

*8.G.C.9:* “Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.”

**Definition of the Topic**

The **total surface area of a cylinder** refers to the area that the surface of the cylinder occupies, precisely the sum of the areas of its two circular bases and the area of its side, which forms a rectangle when unrolled.

**Key Concepts**

** Base Area:** The area of each cylinder base, which is a circle. It is given by the formula A = πr².

** Lateral Area:** The area of the side of the cylinder. It is equal to the height (h) of the cylinder multiplied by the circumference of the base. It is given by the formula A = 2πrh.

** Surface Area:** This is the total area of the cylinder, which is the sum of the base areas and the lateral area.

**Discussion with Illustrative Examples**

To solve for the surface area of a cylinder, we have to get the measures of the area occupied by its curved surface and circular bases.

If a cylinder is flattened into a form like a net, the surface area would equal the sum or total of the areas of all the shapes (a rectangle and two circles) that make up the cylinder.

*Here are the formulas for getting the surface area of a cylinder:*

**Curved Surface Area (C)**

C = 2πrh

**Area of the Circular Bases (B)**

B = 2πr^{2}

**Total Surface Area (T)**

T = 2πrh + 2πr^{2}

*Understanding the Formulas*

A cylinder is composed of a curved surface and two flat circular bases. A cylinder has two types of surface area: ** the curved or lateral surface area** and the

*total surface area.**Curved Surface Area*

The curved surface becomes a rectangle when flattened, with its length as the circumference of the base and its width as the height of the cylinder.

Therefore, to get the curved surface area (C), we multiply the base’s circumference and the cylinder’s height. That is,

Curved Surface Area (C) = 2πrh

*Total Surface Area*

A cylinder’s total surface area is the sum of the total of the areas of its curved surface and the two circular bases. Therefore, to get the total surface area (T), we add the curved surface area and twice the area of a circular base.

Total Surface Area (T) = 2πrh + 2πr^{2}

For both equations,

r = radius of the base

h = height of the cylinder

To understand the concept better, let’s take an example:

Suppose we have a cylinder with a height of 5 cm and a radius of 3 cm.

Using the formula for surface area:

T = 2πrh + 2πr²

T=(2π)(3 cm)(5 cm)+(2π)(3 cm)^{2}

T= 30π cm^{2} + 18π cm^{2}

T= 48π cm²

Thus, the total surface area of the cylinder is **48π square centimetres.**

**Examples with Solution**

**Example 1**

A soup can have a cylinder shape with height = 10 cm and radius = 4 cm. What is its surface area?

**Solution**

T = 2πrh + 2πr²

T=(2π)(4)(10)+(2π)(4)^{2}

T= 80π + 32π

T= 112π

Therefore, the total surface area of the soup can is **112π cm**** ^{2}**.

Remember always to write the unit of measurement on your final answers! Since we’re talking about areas, the answers should be in square units.

**Example 2**

Find the curved or lateral surface area and total surface area of the following cylinder. Use π = 3.14.

**Solution**

Area of the Curved Surface:

C = 2πrh

C =(2)(3.14)(2)(5)

C = 62.8 cm2

Area of the Circular Bases

B=2πr^{2}

B=(2)(3.14)(2)^{2}

B=25.12 cm^{2}

Total Surface Area of the Cylinder

T = C+B

T = 62.8 cm^{2} + 25.12 cm^{2}

T = 87.92 cm^{2}

Therefore, the curved surface area is 62.8 cm^{2}, and the total surface area is **87.92 cm**** ^{2}**.

**Example 3**

Find the cylinder’s curved surface area and total surface. Use π = 3.14.

**Solution**

The radius is one-half of the diameter, so the radius of the base is 10 ÷ 2 = 5 m.

Area of the Curved Surface:

C = 2πrh

C =(2)(3.14)(5)(18)

C = 565.2 m^{2}

Total Surface Area:

T = 2πrh + 2πr^{2}

T = 565.2 m^{2} +(2)(3.14)(5 m)^{2}

T = 565.2 m^{2} + 157 m^{2}

T = 722.2 m^{2}

Therefore, the curved surface area is **565.2 m**** ^{2}**, and the total surface area is

**722.2 m**

^{2}**.**

**Real-life Application with Solution**

Suppose you want to paint a cylindrical water tank with a radius of 2 m and a height of 6 m. If one can of paint covers 10 square meters, how many cans of paint do you need?

**Solution**

First, calculate the surface area of the tank:

T = 2πrh + 2πr²

T=(2π)(2)(6)+(2π)(2)²

T= 24π + 8π

T= 32π m²

Therefore, the surface area of the tank is **32π m**** ^{2}**, which is approximately

**100.53 square meters.**

Now, to find out how many cans of paint you need, divide the surface area by the area that one can of paint covers:

Cans of Paint = Surface Area / Area per Can

Cans of Paint = 100.53 / 10

Cans of Paint = 10.053

Since you can’t buy a fraction of a paint can since it is sold per can, you’d need to round up to the nearest whole number. Thus, you would need **11 cans of paint** to cover the cylindrical water tank fully.

**Practice Test**

A. Complete the table below.

B. Answer the following problem. Use π = 3.14.

1. Calculate the total area of the surfaces of a can of cranberries with a radius of 43 mm and height of 42 mm?

2. A cylindrical pillar has a height of 10 m and a radius of 2 m. What is its surface area?

3. A cylindrical container with a height of 15 cm and a radius of 7 cm is to be covered with plastic wrap. What is the surface area to be covered?

4. The radius of a cylindrical oil drum is 0.9 m, and its height is 1.2 m. What is its surface area?

5A cylindrical water bottle has a radius of 5 cm and a height of 25 cm. What is its surface area?

*Answers:*

A.

1. Radius = 6 cm, Height = 15 cm, Curved Surface Area = 180π cm^{2}, Total Surface Area = 252π cm^{2}

1. Radius = 10 in, Height = 8 in, Curved Surface Area = 160π in^{2}, Total Surface Area = 360π in^{2}

1. Radius = 3.1 ft, Height = 14.8 ft, Curved Surface Area = 91.76π ft^{2}, Total Surface Area = 110.98π cm^{2}

B.

1. 22953.4 mm^{2}

2. 150.72 m^{2}

3. 967.12 cm^{2}

4. 11.8692 m^{2}

5. 942 cm^{2}

**Frequently Asked Questions (FAQs)**

### What is the difference between a cylinder’s lateral area and surface area?

A cylinder’s lateral surface area is the area of the curved surface only, not including the bases. The surface area of a cylinder includes both area of the two circular bases and the lateral area.

### What is the role of π in the formula for the surface area of a cylinder?

“π”, a mathematical constant, is the ratio of any circle’s circumference to its diameter. It is essential in the formula for the surface area of a cylinder because calculating the lateral area and the two bases involves circles.

### Can a cylinder have a surface area of zero?

No, a cylinder cannot have a surface area of zero. Even if the height or the radius is zero, the cylinder would still have a surface area from its base or side, respectively.

### What if the cylinder is not a right circular cylinder?

The formula provided for the surface area is for the right circular cylinders only. A right circular cylinder is one where the bases are perpendicular to its height. If the cylinder is not right, more complex mathematics would be needed to calculate the surface area.

### Do all cylinders, regardless of size, follow the same formula for calculating their surface area?

The formula applies to all right circular cylinders, regardless of their size. The radius and height are variables, which means they can be any real number.

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