RC Reminder Calculator

Cylinder Volume From Surface Area

Know the total surface area of a cylinder? This calculator works backwards to find the radius and then computes the volume. You'll need either the surface area plus the height, or the surface area plus the radius. The tool handles the algebra for you.

Volume From Surface Area

h = (SA − 2πr²) / (2πr), V = πr²h
cm²
Lateral: 2πrh πr² πr² SA = 2πr² + 2πrh → Solve for h, then V = πr²h

Surface Area to Volume Relationship

lateral πr² πr² SA = 2πr² + 2πrh

The total surface area of a cylinder is SA = 2πr² + 2πrh. This includes both circular bases (2πr²) and the curved side (2πrh).

If you know SA and h, you can solve for r using the quadratic formula: 2πr² + 2πhr − SA = 0 r = [−2πh + √((2πh)² + 8π·SA)] / (4π)

Once you have r, plug into V = πr²h for the volume.

When You Know Surface Area and Radius

h = (SA−2πr²) / (2πr)

If you know the surface area and the radius, finding the height is simpler: h = (SA − 2πr²) / (2πr)

Then V = πr²h as usual.

This case arises when you know how much material covers the cylinder (like sheet metal or wrapping paper) and you know the base size. The height is the unknown, and the volume follows directly.

Real-World Uses

Material & packaging

Manufacturers often know the surface area because it determines material cost. A tin can's surface area dictates how much sheet metal is needed. A label's dimensions give the lateral surface area.

Packaging engineers optimize by minimizing surface area for a given volume (to save material) or maximizing volume for a given surface area (to hold more product). The optimal cylinder has height equal to its diameter (h = 2r).

Paint coverage calculations also start with surface area and work backward to dimensions and volume.

Frequently Asked Questions

Can I find the volume from surface area alone?
Not uniquely. Many different cylinders can share the same surface area but have different volumes. You need at least one additional measurement — the height or the radius — to determine the volume.
What is the formula for cylinder surface area?
SA = 2πr² + 2πrh = 2πr(r + h). This includes both circular bases and the curved lateral surface.
What cylinder shape gives the most volume for a given surface area?
A cylinder where h = 2r (height equals diameter). This is the most efficient shape, holding the maximum volume for the minimum material.
How do I find the lateral surface area only?
The lateral (side) surface area is 2πrh. This is the area of the curved surface, excluding the two circular bases.
What is the surface area to volume ratio?
SA/V = (2πr² + 2πrh) / (πr²h) = 2/h + 2/r. Larger cylinders have smaller ratios, which is why big tanks retain heat better than small ones.