Imagine a situation when we have a cylinder and its height is doubled. We are asked to decide if its surface area will also double. Let B be the base of the cylinder and h be its height.
The surface area of a cylinder its lateral area plus the area of the two circular bases.
Surface Area = Lateral Area + 2(Area of Base)
In the graph we can see that changing the height of the cylinder only affects the lateral area of the cylinder — it doubles it. The area of the bases remains unchanged.
For the total area to double, the area of the bases should be twice what it was as well. This is why the total area does not double. Let's take a look at an example.
Example
Here are two cylinders with the radius of the base equal to 5 cm. The height of the first cylinder is 6 cm. The height of the second cylinder is twice the height of the first cylinder, which is 12 cm.
We want to know if the surface area of the second cylinder is twice the surface area of the first cylinder. For that, we need to calculate their surface areas. Let's remember the formula for the surface area of a cylinder.
S.A. = 2π r h + 2π r^2
Here r is the radius of the base and h is the height of the cylinder. We can now substitute the dimensions of each cylinder into the formula and evaluate, starting with the first cylinder.
The surface area of the second cylinder is about 534 cm^2. This is not twice 346 cm^2.
534 cm^2 ≠ 2(346 cm^2)
This means that doubling the height does not double the surface area of a cylinder.
