What is the total surface area of a right circular cone of height $ 14cm $ and base radius $ 7cm $ ?
(A) $ 344.35c{{m}^{2}} $
(B) $ 462c{{m}^{2}} $
(C) $ 498.35c{{m}^{2}} $
(D) None of these

Hint: For answering this question we will use the information given in the question which is stated as “The height and base radius of a right circular cone is $ 14cm $ and $ 7cm $ respectively”. The formulae for the total surface area of a right circular cone is given as $ \pi r\left( l+r \right) $ where $ r $ is the base radius and $ l $ is the slant height of the cone which is given as $ l=\sqrt{{{r}^{2}}+{{h}^{2}}} $ where $ h $ is the height.

Complete step by step answer:
Now considering the question we have a right circular cone with height $ 14cm $ and base radius $ 7cm $ .
From the basic concept we know that the formula of the total surface area of a right circular cone is given as $ \pi r\left( l+r \right) $ where $ r $ is the base radius and $ l $ is the slant height of the cone which is given as $ l=\sqrt{{{r}^{2}}+{{h}^{2}}} $ where $ h $ is the height.
So now we will derive the slant height of this right circular cone using the formulae $ l=\sqrt{{{r}^{2}}+{{h}^{2}}} $ where $ h $ is the height.
From the question, we have been given that, height is $ 14cm $ and base radius is $ 7cm $

Slant height of the cone is given as
 $ \begin{align}
  & l=\sqrt{{{\left( 7cm \right)}^{2}}+{{\left( 14cm \right)}^{2}}} \\
 & \Rightarrow \sqrt{49c{{m}^{2}}+196c{{m}^{2}}} \\
 & \Rightarrow \sqrt{245c{{m}^{2}}} \\
 & \Rightarrow 7\sqrt{5}cm \\
\end{align} $
Therefore, slant height of the right circular cone is $ 7\sqrt{5}cm $
Now we have to calculate the total surface area of the right circular cone for that we will use the formulae which is given as $ \pi r\left( l+r \right) $
By using the formulae we will have,
 $ \begin{align}
  & \text{T}\text{.S}\text{.A of the cone}=3.14\left( 7cm \right)\left[ \left( 7\sqrt{5}cm \right)+\left( 7cm \right) \right] \\
 & \Rightarrow 22\left( 7 \right)\left[ 1+\sqrt{5} \right]c{{m}^{2}} \\
 & \Rightarrow 154\left[ 1+\sqrt{5} \right]c{{m}^{2}} \\
 & \Rightarrow 154\left( 3.236 \right)c{{m}^{2}} \\
 & \Rightarrow 498.35c{{m}^{2}} \\
\end{align} $
Because $ \sqrt{5}=2.236 $ .
Hence we can conclude that the total surface area of the given right circular cone is $ 498.35c{{m}^{2}} $ .
Therefore, option C is the correct option.

Note:
 While answering questions of this type we should be careful while substituting the values in the formula. And calculations must be done very carefully if we commit a single mistake like using the $ \sqrt{5} $ value as $ 1.236 $ it will change the answer completely it will give the answer as $ 344.344c{{m}^{2}} $ which is wrong.