Question

Find the formula for finding the total surface area of a cylinder having a cone shaped lid at both ends.
A. \[\pi r(l + 2r)\]
B. \[\pi r(r + 2h)\]
C. \[2\pi r(l + h)\]
D. \[2\pi r(h + 2r)\]

Answer 1

Hint: We will divide the given solid in three parts, two with cones and 1 with open cylinder at both sides and then find their surface areas individually by assuming the radius to be r and height to be h. Then we will add them all to get our answer.

Complete step-by-step answer:
Let say the radius of the cones and cylindrical base be r units.
We considered them to be equal because it is given that cones are acting as lids to the solid.
Now, let the height of the cylinder be h units.
We know that we need slant height of cones for their surface area. Let that be l units.
We will use c with S for the notation of surface area of cones and C with S for surface area of cylinder.
Now, we also know that surface area of a cylinder is given by:-
${S_C} = 2\pi rh$ , where r is the radius of base of the cylinder and h is the height of the cylinder.
So, we have the surface area of the cylindrical part with us.
${S_C} = 2\pi rh$ ……….(1)
Now, we have two conical shapes with open bottoms.
If we find the surface area of one, we can multiply it with 2, to get the surface area of both the cones which will be
We know that surface area of a cone with open base is given by:-
${S_c} = \pi rl$ , where r is the radius of base of the cone and l is the slant height of the cylinder.
Now, multiplying it by 2, we have surface area of both the cones which will be ${S_c} = 2\pi rl$ ……(2)
Adding both (1) and (2), we will get:-
${S_c} + {S_c} = 2\pi rl + 2\pi rh$
Taking $2\pi r$ common on the RHS, we will get:-
Total surface area required = ${S_c} + {S_c} = 2\pi r(l + h)$.

So, the correct answer is “Option C”.

Note: The students must keep in mind to take general notations for everything like l for slant height, h for height of cylinder and r for the radius of base because if you replace them with something, you will never reach a conclusion from the given options.
You must also remember that it may be given that cones have a base as well. So, we will then need to find out the surface areas of the bases as well. So, mold as per your needs in the question and just remember to divide shapes to solve it easily.