You've provided the condition \( h = 2r \). 

Substituting \( h = 2r \) into the formula for the lateral surface area of the cylinder:

\[
A = 2\pi rh
\]

Now replace \( h \) with \( 2r \):

\[
A = 2\pi r(2r) = 4\pi r^2
\]

So, the formula for the lateral surface area \( A \) in terms of the radius \( r \) alone is:

\[
A = 4\pi r^2
\]

This shows that the lateral surface area of the cylinder depends only on the square of the radius \( r \) when \( h = 2r \).

Do you want to calculate the surface area for a specific radius, or explore further?

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Here are 5 related questions to think about:
1. How does the surface area of a cylinder change as the radius increases?
2. What is the significance of the factor \( 4\pi r^2 \) in the formula?
3. How does the volume of a cylinder change when \( h = 2r \)?
4. How would the formula change if \( h \) was a different multiple of \( r \)?
5. Can you derive the formula for total surface area (including top and bottom) when \( h = 2r \)?

**Tip:** When both the height and radius are proportional (e.g., \( h = 2r \)), the geometry of the cylinder becomes more symmetric, simplifying some calculations like surface area.

Explore how the lateral surface area of a cylinder changes when the height is twice the radius (h = 2r). Learn the formula A = 4πr^2 and its implications. Ideal for students and enthusiasts studying geometry and cylinders.

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