Question

A sphere, cylinder and a cone gave the same radius and same height, find the ratio of their curved surface areas?

Answer 1

Hint: To specify a figure most fundamental quantities are perimeter, area and volume. Area can be defined as the space occupied by a flat shape or the surface of an object. So, using this definition we can easily solve our problem.

Complete step-by-step answer:

Curved surface area of a figure is defined as the area of the curvature of the surface. In this problem, we are given three solids namely sphere, cylinder and cone. All the solids have the same radius and same height. Let the radius of each solid be r and since the height of the sphere is equivalent to diameter, let the height be 2r.
Now, curved surface area of sphere $=4\pi {{r}^{2}}$
Also, curved surface area of cylinder $=2\pi rh=2\pi r(2r)=4\pi {{r}^{2}}$
Finally, the curved surface area of cone $=\pi rl$
Now, to determine the slant height we have,
$\begin{align}
  & l=\sqrt{{{r}^{2}}+{{h}^{2}}} \\
 & l=\sqrt{{{r}^{2}}+{{(2r)}^{2}}} \\
 & l=\sqrt{{{r}^{2}}+4{{r}^{2}}} \\
 & l=\sqrt{5{{r}^{2}}}=r\sqrt{5} \\
\end{align}$
Therefore, the curved surface area of cone $=\pi {{r}^{2}}\sqrt{5}$
Since, all the areas are having only r as variables, we can eliminate this by taking ratio.
Now, for finding the ratio of all the curved surface area we take all the three terms and divide them to get:
$\begin{align}
  & \Rightarrow 4\pi {{r}^{2}}:4\pi {{r}^{2}}:\pi {{r}^{2}}\sqrt{5} \\
 & \Rightarrow 4:4:\sqrt{5} \\
\end{align}$
So, the ratio of curved surface area of sphere, cylinder and cone is $4:4:\sqrt{5}$.

Note: The key step for solving this question is the concept of curved surface area of various solids. By using the formula and putting appropriate values of height and radius, we obtain our solution without any error.