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# Surface Area of a Cylinder – Explanation & Examples

Before we jump into the topic of a cylinder’s surface area, let’s review a cylinder. In geometry, a cylinder is a three-dimensional figure with two circular bases parallel to each other and a curved surface.

## How to Find the Surface Area of a Cylinder?

**The surface area of a cylinder is the sum of two parallel and congruent circular faces and the curved surface area. **

This article will discuss **how to find the total surface area and lateral surface area of a cylinder**.

**To calculate the surface area of a cylinder**, you need to find the Base Area (B) and Curved Surface Area (CSA). Therefore, the surface area or the total surface of a cylinder is equal to the sum of the base area times two and the area of the curved surface.

The curved surface of a cylinder is equal to a rectangle whose length is 2**πr **and whose width is **h.**

Where r = radius of the circular face and h = height of the cylinder.

The area of the curved surface = Area of a rectangle =l x w** = πdh**

The base area, B = Area of a circle** = πr ^{2}**

*The area of a cylinder formula*

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

Total surface area of a cylinder = 2πr^{2} + 2πrh

**TSA = 2πr ^{2} + 2πrh**

Where 2πr^{2} is the top and bottom circular face area, and 2πrh is the area of the curved surface.

By taking 2πr as a common factor from RHS, we get;

**TSA = 2πr (h + r) **……………………………………. (*Surface area of a cylinder formula*)

Let’s solve example problems involving the surface area of a cylinder.

*Example 1*

Find the total surface area of a cylinder whose radius is 5 cm and height is 7 cm.

__Solution__

By the formula,

TSA = 2πr (h + r)

= 2 x 3.14 x 5(7 + 5)

= 31.4 x 12

= 376.8 cm^{2}

*Example 2*

Find the radius of a cylinder whose total surface area is 2136.56 square feet, and the height is 3 feet.

__Solution__

Given:

TSA = 2136.56 square feet

Height, h = 3 feet

But, TSA = 2πr (h + r)

2136.56 =2 x 3.14 x r (3 + r)

2136.56 = 6.28r (3 + r)

By distributive property of multiplication on the RHS, we have,

2136.56 = 18.84r + 6.28r^{2}

Divide each term by 6.28

340.22 = 3r + r^{2}

r^{2} + 3r – 340.22 = 0 ……… (**a quadratic equation**)

By solving the equation using the quadratic formula, we get,

r = 17

Therefore, the radius of the cylinder is 17 feet.

*Example 3*

The cost of painting a cylindrical container is $0.04 per cm^{2}. Find the cost of painting 20 containers of radius, 50 cm, and height, 80 cm.

__Solution__

Calculate the total surface area of 20 containers.

TSA = 2πr (h + r)

= 2 x 3.14 x 50 (80 + 50)

= 314 x 130

= 40820 cm^{2}

The total surface area of 20 containers = 40,820 cm^{2} x 20

=816,400 cm^{2}

The cost of painting = 816,400 cm^{2 }x $0.04 per cm^{2}

= $32,656.

Hence, the cost of painting 20 containers is $32,656.

*Example 4*

Find the height of a cylinder if its total surface area is 2552 in^{2 }and the radius is 14 in.

__Solution__

Given:

TSA = 2552 in^{2}

Radius, r = 14 in.

But, TSA = 2πr (h + r)

2552 = 2 x 3.14 x 14 (14 + h)

2552 = 87.92(14 + h)

Divide both sides by 87.92 to get,

29.026 = 14 + h

Subtract by 14 on both sides.

h = 15

Hence, the height of the cylinder is 15 in.

## Lateral Surface Area of a Cylinder

**As stated before, the area of the curved surface of a cylinder is what is termed as the lateral surface area. In simple words, a cylinder’s lateral surface area is the surface area of a cylinder, excluding the area of the base and bottom (circular surface).**

The formula gives the lateral surface area of a cylinder;

**LSA = 2πrh**

*Example 5*

Find the later surface area of a cylinder whose diameter is 56 cm and height is 20 cm.

__Solution__

Given:

Diameter = 56 cm, hence radius, r =56/2 = 28 cm

Height, h = 20 cm

By, the formula,

LSA = 2πrh

= 2 x 3.14 x 28 x 20

= 3516.8 cm^{2}.

Thus, the lateral surface area of the cylinder is 3516.8 cm^{2}.

*Example 6*

The lateral surface area of a cylinder is 144 ft^{2}. If the radius of the cylinder is 7 ft, find the height of the cylinder.

__Solution__

Given;

LSA = 144 ft^{2}

Radius, r = 7 ft

144 = 2 x 3.14 x 7 x h

144 = 43.96h

Divide by 43.96 on both sides.

3.28 = h

So, the height of the cylinder is 3.28 ft.