What is a Cylinder?
Two circular bases of equal size make up a cylinder. A cylinder’s height h is the distance between its two bases. And the height, h, of all the cylinders we’ll work with here, will be perpendicular to the bases.
The area of all the faces of the cylinder is the surface area of the cylinder. The area of all its faces excluding the area of the bases is referred to as the lateral surface area.
Formula to find the Surface area of a Cylinder
Let’s consider a can to grasp the formula for a cylinder’s surface area. The can has a height h and a radius r. We can also assume that the can has a label covering it except the bottom and top parts of the tin.
We can see that the label is actually a rectangle by carefully cutting the label and unraveling it. The rectangle’s length equals the circumference of the cylinder’s base,”L” and its width “W” equals the cylinder’s height.
As a result, the label’s area is the lateral surface area of the cylinder. This can be represented as \(A~=~l\times~w\) where, \(L~=~2\pi~r\) and \(w~=~h\).
Hence, the lateral surface area of a cylinder is \(2\times~\pi\times~r\times~h\), where \(r\) is the base radius and h is the cylinder’s height.
We add the areas of the two circles to the area of the rectangle to get the surface area of the cylinder.
\(S~=~A_{\text{top circle}}~+~A_{\text{bottom circle}}~+~A_{\text{rectangle}}\)
\(S~=~\pi~r^2+~\pi~r^2+2\pi~rh\times~h\)
\(S~=~2\times~\pi~r^2+~2~\pi~rh\)
\(S~=~2\pi~r^2~+~2\pi~rh\)
A cylinder with radius r and height h has a surface area of \(S~=~2\pi~r^2+~\pi~rh\)
Solved Examples
1) Calculate the surface area of the cylinder having a radius of 20 centimeters and a height of 30 centimeters.
Solution:
Given data,
\(r~=~20\) centimeters
\(h~=~30\) centimeters
Let “S” be the surface area of the cylinder.
As we know, a cylinder with radius r and height h has a surface area of \(S~=~2\pi~r^2~+~2\pi~rh\)
\(S~=~2\pi~(20)^2~+~2\pi~(20)(30)\)
\(S~=~6283.19\) square centimeters
Therefore, the surface area of the cylinder is 6283.19 square centimeters.
2) A can of black coffee has a 6 cm radius and a 12 cm height. Determine its surface area.
Solution:
Given data,
\(r~=~6\) centimeters centimeters
\(r~=~12\) centimeters centimeters
Let “S” be the surface area of the cylinder.
As we know, a cylinder with radius r and height h has a surface area of \(S~=~2\pi~r^2~+~2\pi~rh\)
\(S~=~2\pi~(6)^2~+~2\pi~(6)(12)\)
\(S~=~678.58\) square centimeters
Therefore, the surface area of the can of black coffee is 678.58 square centimeters.
3) The diameter of a cylindrical architectural column is 4 feet, and the height is 21 feet. Calculate the column’s surface area.
Solution:
Given data,
d = 6 feet
We know that diameter \((d)~=~2r\)
\(r~=~\frac{d}{2}\)
\(r~=~\frac{6}{2}\)
\(r~=~3\) feet
\(r~=~21\) feet
Let “S’ be the surface area of the cylinder.
As we know, a cylinder with radius r and height h has a surface area of \(S~=~2\pi~r^2~+~2\pi~rh\)
\(S~=~2\pi~(3)^2~+~2\pi~(3)(21)\)
\(S~=~452.39\) square feet
Therefore, the surface area of the can of a cylindrical architectural column is 452.39 square feet.
3) A snack pack of cookies has a cylindrical shape with a radius of 6 cm and a height of 5 cm. Determine the lateral surface area.
Solution:
Given data,
\(r~=~6\) centimeters
\(h~=~5\) centimeters
Let ‘l’ be the lateral surface area of the cylinder.
As we know, a cylinder with radius r and height h has a lateral surface area of \(S~=~2\pi~rh\)
\(l~=~2\pi~(6)(5)\)
\(l~=~188.5\) square centimeters
Therefore, the lateral surface area of the snack pack of cookies is 188.5 sq.centimeters.
Surface area of a cylinder \(=~2\pi~r(r+h)\), where the radius is “r’”, and the height of the cylinder is “h”. The area of the two bases (\(2\pi~r^2\) each) and the lateral surface area \((2\pi~rh)\) make up the surface area.
The formula to calculate the curved or lateral surface area of a cylinder is \(2\pi~rh\), where “r” is the radius and “h” is the cylinder’s height.
The area of one base and the area of the curved surface can be used to calculate the surface area of a cylinder with an open top. The surface area of an open-top cylinder \(=~\pi~r^2~+~2\pi~rh~=~\pi~r(r+2h)\), where “r” is the radius and “h” is the height of the cylinder. Because the cylinder does not have a top, we have taken the area of just one base.
A hollow cylinder is open from the top and bottom. Hence the surface area of a hollow cylinder is actually the lateral surface area of that cylinder.
The surface area of a cylinder is the sum of areas of all its faces, whereas the lateral surface area is the sum of areas of all its faces except the bases.