Curved Surface Area of a Cylinder (2πrh) Explained

Grades 8-9

Question: pi x 20 x 40

The expression

\pi \times 20 \times 40

represents a geometric calculation typically associated with the lateral surface area of a cylinder.

To understand this visually, imagine a cylinder with a diameter of

20

units and a height of

40

units. The calculation can be broken down into two parts:

The Circumference ( $\pi \times 20$ ): The value $20\pi$ represents the distance around the circular base of the cylinder. If you were to "cut" the cylinder vertically and roll it out flat, this circumference would become the width of a rectangle.
The Height ( $40$ ): This is the vertical length of the cylinder. In our "unrolled" rectangle, this becomes the length.

The area

A

of the curved surface is found by multiplying the perimeter of the base by the height:

\begin{align*} \text{Area} &= \text{Circumference} \times \text{Height} \\ \text{Area} &= (\pi \times \text{diameter}) \times \text{height} \\ \text{Area} &= \pi \times 20 \times 40 \\ \text{Area} &= 800\pi \\ \text{Area} &\approx 2513.27 \text{ square units} \end{align*}

In the interactive diagram below:

The blue circle represents the base of the cylinder with a diameter of $20$ (radius of $10$ ).
The vertical segment represents the height of $40$ .
The gray shaded region represents the lateral surface whose area is exactly $800\pi$ .
You can use the Rotation slider to view the cylinder from different perspectives.