Question

A cylinder with radius r and height h is closed on the top and bottom. Which of the following expressions represents the total surface area of this cylinder?
A. $2\pi r\left( {r + h} \right)$
B. $\pi r\left( {r + 2h} \right)$
C. $\pi r\left( {2r + h} \right)$
D. $2\pi {r^2} + h$

Answer 1

Hint:
A cylinder contains a curved surface and two circular bases so we can write-
The total surface area of cylinder=curved surface area of cylinder + area of the two circular bases
For area of curved surface use the formula of area of rectangle = L× B because area of the rectangle is equal to the area of the curved surface where the length of the cylinder (L) is equal to the circumference of the circular base and the breadth B is the height of the cylinder. For the area of circular bases use the formula of area of circle=$\pi {r^2}$ where r is the radius of the circle. Put all the obtained values in the expression of total surface area of the cylinder to get the answer.

Complete step by step solution:
A cylinder contains a curved surface and two circular bases. Let the height of the curved surface be h. and the radius of the circular bases is r. Now we have to determine the expression that gives us the total surface area of the cylinder.
So we write that,
 The total surface area of cylinder=curved surface area of cylinder + area of the two circular bases-- (i)

Then from the figure we can say that the length of the cylinder (L) is equal to the circumference of the circular base
$ \Rightarrow L = 2\pi r$ {Because the circumference of a circle with radius r=$2\pi r$ }
And the breadth B is the height of the cylinder
$ \Rightarrow B = h$

Then we know that area of the rectangle= area of the curved surface
Then Area of curved surface =L× B
On putting the values we get,
Area of curved surface=$2\pi rh$ --- (ii)
Now we have to calculate the area of the circular base
We know that the area of circle is given as-

Area of circle=$\pi {r^2}$ where r is the radius of the circle
Then the area of the two circular bases=$\pi {r^2} + \pi {r^2}$
On solving we get,
The area of two circular bases=$2\pi {r^2}$ -- (iii)
On substituting the values of eq. (ii) and eq. (iii) in eq. (i), we get,
The total surface area of cylinder=$2\pi rh + 2\pi {r^2}$
On taking $2\pi r$ common from both terms,
Total surface area of cylinder=$2\pi r\left( {h + r} \right)$

Hence the correct answer is A.

Note:
Here the student may go wrong if they only take area of one circular base to add in the expression of the total surface area considering that the area of both circular bases will be same because then the total surface area of the cylinder will become-$\pi r\left( {h + 2r} \right)$ which is wrong expression . Since there are two circular bases so the area of both circular bases has to be added in the expression to get total surface area.