Surface of a cylinder

The surface area of a cylinder tells us how big is the area of the surfaces that bound the cylinder. Careful not to confuse this with the volume of the cylinder, which tells us "how many litres of water we are able to pour into the cylinder". Let's look at what such a cylinder looks like:

Formula

If you are just looking for a formula, then the surface area of a cylinder whose base has a radius of r and which has a height of v, is calculated as

$$\Large S=2\cdot\pi\cdot r^2+2\cdot\pi\cdot r \cdot v$$

Calculator: calculate the surface area of a cylinder

How did we come up with the formula?

Every cylinder has two bases: these are the two circles at the top and bottom. Next, the cylinder has a shell, which is the area "between" the bases. If we want to calculate the surface of the cylinder, we need to calculate the area of the two bases, the area of the shell, and add these values together. Since the base is made up of a circle, we use the formula to calculate the area of the circle

$$\Large S_\circ=\pi\cdot r^2,$$

where r is the radius of the base. In the figure, that's the horizontal dashed line. For the example, if r = 6, then the area of the base is equal to

$$\Large S_\circ=\pi\cdot 6^2 = 36\pi$$

Next, we need to calculate the area of the shell. We can notice that the shell is formed by a "rolled up" rectangle. One side of the rectangle has a length equal to the height v of the cylinder, the vertical dashed line in the figure. The other side, when "unrolled", would be the same size as the circumference of the circle that forms the base. So we calculate the circumference of the base first, using the formula for calculating the circumference of a circle, which is

$$\Large o = 2\cdot\pi\cdot r$$

After substitution, we get:

$$\Large o = 2\cdot\pi\cdot 6 = 12 \pi$$

The perimeter of the base is 12π. We calculate the volume of the rectangle, let's label it $S_\square$, simply by multiplying the lengths of the two sides:

$$\Large S_\square = 12\pi \cdot v$$

If the height of the cylinder were, say, v = 10, we would get

$$\Large S_\square = 12\pi \cdot 10 = 120\pi$$

The total surface area of the cylinder, let's call it S, would be obtained by adding all the results (and remembering that the cylinder has two bases):

$$\Large S=S_\circ+S_\square+S_\circ$$

That is, after homing:

$$\Large S=36\pi+120\pi+36\pi=192\pi$$

If you don't like the result with the constant π, you can substitute the approximate value 3,14 instead of π to get the approximate result

$$\Large S\approx 192\cdot3{,}14=602{,}88$$