Last updated: 2026-05-18

Cylinder volume and surface area

A right circular cylinder is a solid with two identical flat circular ends, called bases, joined by a curved surface, where the central axis is perpendicular to both bases. It is defined by two measurements: the base radius r and the height h. From these, three quantities follow — the volume, the lateral (curved) surface area, and the total surface area.

The formulas

For a cylinder with base radius r and height h:

Property	Formula
Volume	V = πr²h
Lateral surface area	A_lat = 2πrh
Total surface area	A_tot = 2πr(r + h)

Where the formulas come from

The volume is the area of the circular base (πr²) multiplied by the height h. The cylinder can be treated as a stack of infinitely thin circular disks from bottom to top, each of area πr²; summing them over the height gives πr²h. (The covers the area and circumference of the base alone.)

The lateral surface area follows from unrolling the curved wall. Cutting it vertically and laying it flat produces a rectangle whose width equals the base circumference (2πr) and whose height equals h, giving an area of 2πrh.

The total surface area adds the two circular caps (each πr²) to the lateral area: 2πrh + 2πr² = 2πr(r + h).

Worked example

A water storage tank has a base radius r = 0.5 m and height h = 1.2 m:

• Volume = π × 0.25 × 1.2 ≈ 0.942 m³ = 942 litres
• Lateral area = 2π × 0.5 × 1.2 ≈ 3.77 m² (the sheet metal needed for the wall)
• Total area = 2π × 0.5 × (0.5 + 1.2) ≈ 5.34 m² (including the two circular caps)

Unit conversion: cm³ to litres and m³

Volume in cubic centimetres, litres, and cubic metres are directly related:

• 1 L = 1000 cm³ — divide cm³ by 1000 to get litres
• 1 m³ = 1000 L — a cubic metre holds 1000 litres
• 1 m³ = 1,000,000 cm³

A standard 330 mL drink can has radius ≈ 3.3 cm and height ≈ 9.6 cm: V = π × 3.3² × 9.6 ≈ 328.4 cm³ ≈ 330 mL — matching the nominal label.

Cylinders compared with cones and spheres

For shapes with the same radius and height:

Shape	Volume
Cylinder	πr²h
Cone	(1/3)πr²h
Hemisphere (h = r)	(2/3)πr³

A holds exactly one-third as much as the cylinder with the same base and height — a result proved by Euclid. A of radius r inscribed in a cylinder of radius r and height 2r holds exactly two-thirds of the cylinder's volume (Archimedes' Hat-Box theorem).

Applications

The same three quantities cover most practical cases:

Drinking glass (r = 4 cm, h = 12 cm):

• Volume = π × 16 × 12 ≈ 603 cm³ ≈ 603 mL (about 600 mL, a standard large glass)

PVC pipe (inner r = 5 cm, h = 2 m):

• Volume = π × 0.0025 × 2 ≈ 0.01571 m³ = 15.71 litres of water it can carry

When only the circumference C of a physical cylinder is known (measured with a tape measure), the radius is recovered with:

r = C ÷ (2π)

A can with circumference 20.7 cm has r = 20.7 ÷ (2π) ≈ 3.3 cm. Digital calipers give the most accurate reading when the diameter is measured directly across the base; dividing by 2 gives the radius.

Frequently Asked Questions (FAQ)

How do you calculate the volume of a cylinder?

The volume of a right circular cylinder is V = πr²h, where r is the base radius and h is the height. For example, a cylinder with r = 5 cm and h = 10 cm has V = π × 25 × 10 ≈ 785.4 cm³.

How do you convert cubic centimetres to litres?

1 litre = 1000 cm³, so divide the volume in cm³ by 1000 to get litres. For example, 785.4 cm³ ÷ 1000 = 0.7854 L. Similarly, 1 m³ = 1000 L, and 1 L = 0.001 m³.

How do you find the surface area of a cylinder?

The total surface area has two parts: the lateral (curved) surface and the two circular ends. Lateral area = 2πrh. End area = 2πr². Total surface area = 2πr(r + h). For r = 5 cm and h = 10 cm, total area = 2π × 5 × (5 + 10) = 150π ≈ 471.2 cm².

How do you measure the radius of a cylinder?

Measure the diameter across the widest point of the circular base and divide by 2 to get the radius. For pipes and cans, wrap a tape measure around the circumference and use r = C ÷ (2π). Digital calipers give the most accurate reading for small cylinders.