The volume and surface area formulas
V = πr²h
A = 2πr² + 2πrh
Where the formula comes from
Stack an infinite number of identical circles and you get a cylinder, which is why the volume is the base area πr² multiplied by the height. For the surface, cut the side vertically and unroll it: you get a rectangle whose width is the circumference 2πr and whose height is h, so the lateral area is 2πrh. Add the two circular ends and you have the total.
A worked example, step by step
A cylinder with a base radius of 3 cm and a height of 10 cm
- Base area: π × 3² = 28.27 cm².
- Volume: 28.27 × 10 = 282.74 cm³.
- Lateral surface: 2π × 3 × 10 = 188.50 cm².
- Total surface: 188.50 + 2 × 28.27 = 245.04 cm².
Watch the units: volume goes in cubic units and surface area in square ones. Mixing them is the most common mistake.
Where it turns up in practice
Tanks, pipes, tins and silos are all cylinders, which makes this the most-used volume formula in engineering after the box. It is also how a cylinder engine gets its displacement: bore radius squared, times π, times stroke, times the number of cylinders.