Question

Write the formula for TSA of cylinder.

Answer 1

Hint: We have the formula for the total surface area of any figure is given as
\[TSA=\left( \text{Height} \right)\left( \text{Base perimeter} \right)+\left( \text{Base area} \right)+\left( \text{Top area} \right)\]
We use this formula for the cylinder to find the TSA of any cylinder assuming the required dimensions for the cylinder.

Complete step by step answer:
We are asked to write the TSA of the cylinder.
Let us assume a cylinder of base radius \[r\] and height \[h\] as follows

We know that the base and top of the cylinder is a circle.
We know that the formula of the perimeter of a circle having the radius \[r\] is given as
\[C=2\pi r\]
We also know that the formula of area of a circle having the radius \[r\] is given as
\[A=\pi {{r}^{2}}\]
Now, let us find the total surface area of the cylinder.
We know that the formula for total surface area for any figure is given as
\[TSA=\left( \text{Height} \right)\left( \text{Base perimeter} \right)+\left( \text{Base area} \right)+\left( \text{Top area} \right)\]
Here, we can see that the base and top of the cylinder is a circle.
Let us assume that the total surface area of the cylinder as \[TSA\]
Now, by using the above formula of total surface area of any figure to cylinder then we get
\[\begin{align}
  & \Rightarrow TSA=\left( h \right)\left( 2\pi r \right)+\left( \pi {{r}^{2}} \right)+\left( \pi {{r}^{2}} \right) \\
 & \Rightarrow TSA=2\pi rh+2\pi {{r}^{2}} \\
\end{align}\]
Therefore, we can conclude that the total surface area of the cylinder is given as
\[TSA=2\pi rh+2\pi {{r}^{2}}\]
Where, \[r\] is the base radius of the cylinder and \[h\] is the height of the cylinder.

Note:
Students may do mistake in taking the general formula of total surface area of any 3D figure.
We have the formula for the total surface area of any figure is given as
\[TSA=\left( \text{Height} \right)\left( \text{Base perimeter} \right)+\left( \text{Base area} \right)+\left( \text{Top area} \right)\]
Here, we can see that there are two areas of the base area and the top area.
But students may take the formula as
\[TSA=\left( \text{Height} \right)\left( \text{Base perimeter} \right)+2\left( \text{Base area} \right)\]
This may be correct for the cylinder but there are some figures where there will be no top surface like a cone. In that case, there will be no twice the area of the base.