The formula
A right circular cylinder is defined by its base radius r and height h. Everything comes from those two numbers and π:
The volume is the area of the circular base, πr², stretched up by the height. The lateral surface area is the side wall unrolled into a rectangle of width 2πr and height h. The total surface area adds the two circular end caps (each πr²) to that side wall, which factors neatly into 2πr(r + h).
Worked example
For a cylinder with radius 3 and height 10:
Where this comes up
Cylinder volume is the everyday geometry behind cans, pipes, tanks and silos — anything you need to fill or estimate the capacity of. Surface area matters for material and paint estimates: the lateral area tells you how much label or wrap covers the side, while the total area includes the lids. Because volume scales with the radius squared, a small change in radius affects capacity far more than the same change in height.