Surface area of a cylinder calculator

To find the surface area of a cylinder, add its two ends to its curved side: A = 2πr² + 2πrh, which factors to 2πr(r + h). The first term is the two circular ends, each πr². The second is the curved side. For a radius of 4 cm and a height of 10 cm: the ends are 2 × π × 16 = 32π, the side is 2 × π × 4 × 10 = 80π, and the total is 112π, about 351.858 cm². The result above leads with that total, shows the curved side on its own in a separate row, and gives the volume, 160π cm³, alongside.

The side formula comes from unrolling it. Cut a paper label straight down and flatten it, and it is a rectangle. One edge is the height, 10 cm. The other edge went once around the cylinder, so it is the circumference, 2πr = 8π, about 25.13 cm. The rectangle's area is 8π × 10 = 80π cm², which is 2πrh. That is the lateral area: what a label, a pipe wrap or a coat of paint on the side covers.

Which total you want depends on the object. A closed can or tank has both ends: 2πr² + 2πrh. An open cup or a tank with no lid has one: πr² + 2πrh, here 16π + 80π = 96π, about 301.59 cm². A pipe or a tube has neither, so it is only the lateral area. Most wrong answers to word problems come from counting the wrong number of ends.

The other common mistakes are the usual circle ones. Using the diameter as r makes the end areas four times too big and the side twice too big. And this is an area, in square units: centimeters in, square centimeters out. The volume of the same cylinder is in cubic centimeters, πr²h = 160π, about 502.65 cm³, and it answers a different question: how much it holds, not how much material it takes.

Surface area grows with the square of the size, so a cylinder twice as wide and twice as tall needs four times the material but holds eight times as much. That is why large tanks are cheaper per liter. For a fixed volume, the closed cylinder with the least surface has its height equal to its diameter.

For the shapes next to it: a cone's surface is πr² + πrs, with s the slant height. A sphere's surface is 4πr², which Archimedes showed is exactly the lateral area of the cylinder that just contains it (radius r and height 2r, so 2πr × 2r). A hemisphere is 3πr² with its flat face. All of them are shapes on this calculator.