# Surface Area of a Cylinder

A cylinder can be defined as a solid figure that is bound by a curved surface and two flat surfaces. The surface area of a cylinder can be found by breaking it down into 2 parts:

1. The two circles that make up the caps of the cylinder.

2. The side of the cylinder, which when "unrolled" is a rectangle.

The area of each end cap can be found from the radius r of the circle, which is given by:

*A = πr*^{2}

Thus the total area of the caps is 2πr^{2}.

The area of a rectangle is given by:

*A = height × width*

The width is the height *h* of the cylinder, and the length is the distance around the end circles, or in other words the perimeter/circumference of the base/top circle and is given by:

*P = 2πr *

Thus the rectangle's area is rewritten as:

*A = 2πr × h*

Combining these parts together we will have the total surface area of a cylinder, and the final formula is given by:

*A = 2πr*^{2} + 2πrh

where:

π is Pi, approximately 3.142

r is the radius of the cylinder

h height of the cylinder

By factoring 2πr from each term we can simplify the formula to:

*A = 2πr(r + h)*

The lateral surface area of a cylinder is simply given by: *LSA = 2πr × h*.

**Example 1**: Find the surface area of a cylinder with a radius of 4 cm, and a height of 3 cm.

Solution:

*SA = 2 × π × r*^{2} + 2 × π × r × h

*SA = 2 × 3.14 × 4*^{2} + 2 × 3.14 × 4 × 3

*SA = 6.28 × 16 + 6.28 × 12*

*SA = 100.48 + 75.36*

*SA = 175.84*

Surface area = 175.84 cm^{2}

**Example 2**: Find the surface area of the cylinder with a radius of 5.5cm and height of 10cm.

Solution:

The radius of cylinder = 5.5 cm.

The height of cylinder = 10 cm.

The total surface area of the cylinder is therefore:

*TSA = 2πr(r+h)*

*TSA = 11π (5.5+10)*

*TSA = 170.5 π*

*TSA = 535.6 cm*^{2}

**Example 3**: Find the total surface area of a cylindrical tin of radius 17 cm and height 3 cm.

Solution:

Again as in the previous example:

*TSA = 2πr(r+h)*

*TSA = 2π× 17(17+3)*

*TSA = 2π×17×20*

*TSA = 2136.56 cm*^{2}

**Example 4**: Find the surface area of the cylinder with radius of 6 cm and height of 9 cm.

Solution:

The radius of cylinder: *r = 6 cm*

The height of cylinder: *h = 9 cm*

Total surface area of cylinder is therefore:

*TSA = 2πr(r + h)*

*TSA = 12π (6+9)*

*TSA = 180 π*

*TSA = 565.56 cm*^{2}

**Example 5**: Find the radius of cylinder whose lateral surface area is 150 cm^{2} and its height is 9 cm.

Solution:

Lateral surface area of cylinder is given by:

*LSA = 2πrh*

Given that:

*LSA = 150cm*^{2}

*h = 9cm*

π is the constant and its value = 3.14

Substitute the values in the formula and find the value of r by isolating it from the equation:

*LSA = 2πrh*

150 = 2*× π × r × 9*

*r = 150 / (2×9× π)*

*r = 2.65cm*

So the radius of the cylinder is 2.65 cm.

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