Formula and Method for the Radius of a Cylinder
A cylinder is a solid with two parallel circular bases connected by a curved lateral surface. Its volume follows V = πr²h, where r is the radius of the circular base and h is the height. This calculator solves that equation for the radius, then derives the diameter, circumference, and total surface area from it.
How the calculation works
Starting from V = πr²h, divide both sides by πh to isolate r²: r² = V / (πh). Taking the square root gives the radius: r = √(V / (πh)). Once the radius is known, the diameter is simply d = 2r, the circumference of the circular cross-section is C = 2πr, and the total surface area — the two circular ends plus the curved side — is SA = 2πr² + 2πrh. Enter volume and height in the same unit system; the radius, diameter, and circumference are reported in that linear unit, and the surface area in that unit squared.
Common mistakes
- Units: volume must be entered in cubic units (in³, cm³, m³, etc.) that match the linear unit used for height — mixing units (e.g., a volume in liters with a height in inches) produces an incorrect radius.
- Radius vs. diameter: the volume formula V = πr²h uses radius, not diameter. Reporting the diameter as if it were the radius (or vice versa) doubles or halves the error.
- Lateral vs. total surface area: the lateral (side) surface area alone is 2πrh; the total surface area adds the two circular ends (2πr²). Confusing the two under- or over-estimates material needs.
Real-world applications
- Tank, drum, and canister design: given a target volume and a height limited by available space, find the radius that will hold that volume.
- Plumbing and HVAC: sizing pipes or ducts to move a required volume of fluid or air over a given length.
- Manufacturing: determining the stock radius needed to machine or mold a cylindrical part of a specified volume and length.
- Packaging: sizing cans, tubes, and bottles so a target fill volume fits within a fixed height.